Solve Each Inequality. Then Graph the Solution Set on a Number Line. 1. X
5-1 Solving Inequalities by Addition and Subtraction
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
n
eSolutions Manual - Powered by Cognero Page 1 Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p 5-1 Solving Inequalities by Addition and Subtraction
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
eSolutions Manual - Powered by Cognero Page 2
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}. 5-1 Solving Inequalities by Addition and Subtraction
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3. eSolutions Manual - Powered by Cognero Page 3
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
5-1 SolvingThe inequality Inequalities is true bywhen Addition n is greater and thanSubtraction or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p eSolutions Manual - Powered by Cognero Page 4
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}. 5-1 Solving Inequalities by Addition and Subtraction
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c eSolutions Manual - Powered by Cognero Page 5
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}. 5-1 Solving Inequalities by Addition and Subtraction
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
eSolutions 7 >Manual 20 + c- Powered by Cognero Page 6
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r 5-1 Solving Inequalities by Addition and Subtraction
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
eSolutionsThe Manualsolution- Powered set is {byy |Cognero y 6}. Page 7
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}. 5-1 Solving Inequalities by Addition and Subtraction
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
eSolutions Manual - Powered by Cognero Page 8
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}. 5-1 Solving Inequalities by Addition and Subtraction
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
eSolutions Manual - Powered by Cognero Page 9
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
5-1 SolvingThe inequality Inequalities is true bywhen Addition n is greater and thanSubtraction 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12. eSolutions Manual - Powered by Cognero Page 10
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
5-1 Solving Inequalities by Addition and Subtraction The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29. eSolutions Manual - Powered by Cognero Page 11
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
5-1 Solving Inequalities by Addition and Subtraction The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b eSolutions Manual - Powered by Cognero Page 12 A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
5-1 Solving Inequalities by Addition and Subtraction The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If eSolutionsthereManual is one- Poweredgame left,by Cognero how many points must she score to reach her goal? Page 13 Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
5-1 Solving Inequalities by Addition and Subtraction The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + ( eSolutions Manual - Powered by Cognero Page 14
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
5-1 SolvingThe solution Inequalities set is by Addition and. So, Subtraction Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
eSolutions Manual - Powered by Cognero Page 15
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks. 5-1 Solving Inequalities by Addition and Subtraction
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
eSolutions Manual - Powered by Cognero Page 16 The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks. 5-1 Solving Inequalities by Addition and Subtraction
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
eSolutions Manual - Powered by Cognero Page 17
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
5-1 Solving Inequalities by Addition and Subtraction
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. eSolutions Manual - Powered by Cognero Page 18 a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks. 5-1 Solving Inequalities by Addition and Subtraction
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
eSolutions Manual - Powered by Cognero Page 19 So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x 5-1 Solving If a true Inequalities inequality byis multipliedAddition byand a positiveSubtraction number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a eSolutions Manual - Powered by Cognero Page 20 In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
5-1 Solving Inequalities by Addition and Subtraction So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1). eSolutions Manual - Powered by Cognero Page 21
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7.
5-1 Solving Inequalities by Addition and Subtraction The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5? eSolutions Manual - Powered by Cognero Page 22 Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
5-1 Solvingetc. Inequalities by Addition and Subtraction So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
eSolutions Manual - Powered by Cognero Page 23
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
5-1 Solving Inequalities by Addition and Subtraction So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
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Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
5-1 Solving Inequalities by Addition and Subtraction
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form. eSolutions Manual - Powered by Cognero Page 25
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
5-1 Solving Inequalities by Addition and Subtraction
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
eSolutions Manual - Powered by Cognero Page 26
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
5-1 Solving Inequalities by Addition and Subtraction
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
eSolutions Manual - Powered by Cognero Page 27
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
5-1 Solving Inequalities by Addition and Subtraction
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50 eSolutions Manual - Powered by Cognero Page 28
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
5-1 Solving Inequalities by Addition and Subtraction
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120 eSolutions Manual - Powered by Cognero Page 29
3a = 21
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
5-1 Solving Inequalities by Addition and Subtraction
4p = 120
3a = 21
eSolutions Manual - Powered by Cognero Page 30
Solve each inequality. Then graph the solution set on a number line. x 3 > 7
The solution set is {x | x > 10}.
y
The solution set is {y | y 2}.
g + 6 < 2
The solution set is {g | g < 4}.
p + 4
The solution set is {p | p
10 > n 1
The solution set is {n | n < 11}.
k + 24 > 5
The solution set is {k | k > 29}.
8r + 6 < 9r
The solution set is {r | r > 6}.
8n n 3
The solution set is {n | n 3}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number increased by 4 is at least 10 more than the number. Let n = the number.
The solution set is {n | n }. To check substitute three different values into the original inequality: 6, a number less than 6, and a number greater than 6.
The inequality is true when n is greater than or equal to 6, so the solution checks.
Three more than a number is less than twice the number. Let n = the number.
The solution set is {n | n > 3}. To check substitute three different values into the original inequality: 3, a number less than 3, and a number greater than 3.
The inequality is true when n is greater than 3, so the solution checks.
A thrill ride swings passengers back and forth, a little higher each time up to 137 feet. Suppose the height of the swing after 30 seconds is 45 feet. How much higher will the ride swing?
The ride will swing no higher than 92 more feet.
Solve each inequality. Then graph the solution set on a number line. m 4 < 3
The solution set is {m | m < 7}.
p
The solution set is {p | p
r
The solution set is {r | r
t 3 > 8
The solution set is {t | t > 5}.
b
The solution set is {b | b
13 > 18 + r
The solution set is {r | r < 5}.
5 + c
The solution set is {c | c 4}.
q 30
The solution set is {q | q
11 + m
The solution set is {m | m
h 26 < 4
The solution set is {h | h < 30}.
r 14
The solution set is {r | r
7 > 20 + c
The solution set is {c | c < 27}.
2a 4 + a
The solution set is {a | a 4}.
z z
The solution set is {z | z
w w
The solution set is {w | w 5}.
3y y
The solution set is {y | y 6}.
6x x
The solution set is {x | x
9 + 2a < 3a
The solution set is {a | a > 9}.
Define a variable, write an inequality, and solve each problem. Check your solution. Twice a number is more than the sum of that number and 9. Let n = the number.
The solution set is {n | n >9}. To check substitute three different values into the original inequality: 9, a number less than 9, and a number greater than 9.
The inequality is true when n is greater than 9, so the solution checks.
The sum of twice a number and 5 is at most 3 less than the number. Let n = the number.
The solution set is {n | n 8}. To check substitute three different values into the original inequality: 8, a number less than 8, and a number greater than 8.
The inequality is true when n is equal to or less than 8, so the solution checks.
The sum of three times a number and 4 is at least twice the number plus 8. Let n = the number.
The solution set is {n | n To check substitute three different values into the original inequality: 12, a number less than 12, and a number greater than 12.
The inequality is true when n is greater than or equal to 12, so the solution checks.
Six times a number decreased by 8 is less than five times the number plus 21. Let n = the number.
The solution set is {n | n < 29}. To check substitute three different values into the original inequality: 29, a number less than 29, and a number greater than 29.
The inequality is true when n is less than 29, so the solution checks.
Define a variable, write an inequality, and solve each problem. Then interpret your solution. Keisha is babysitting at $8 per hour to earn money for a car. So far she has saved $1300. The car that Keisha wants to buy costs at least $5440. How much money does Keisha still need to earn to buy the car? Let b = the amount of money that Keisha still needs.
The solution set is {b | b
A recent survey found that more than 21 million people between the ages of 12 and 17 use the Internet. Of those, about 16 million said they use the Internet at school. How many teens that are online do not use the Internet at school? Let n = the number, in millions, of online teens that do not use the Internet at school.
The solution set is {n | n > 5}. So, at least 5 million teens use the Internet somewhere other than school.
A DJ added 20 more songs to his digital media player, making the total more than 61. How many songs were originally on the player? Let n = the original number of songs on the player.
The solution set is {n | n > 41}. So, there were originally more than 41 songs on the digital media player.
Let t = the water temperature in the morning.
The solution set is {t | t
A player s goal was to score at least 150 points this season. So far, she has scored 123 points. If there is one game left, how many points must she score to reach her goal? Let p = the number of points the basket ball player must score to reach her goal.
The solution set is {p | p
Samantha received a $75 gift card for a local day spa for her birthday. She plans to get a hair cut and a manicure. How much money will be left on her gift card after her visit? Service Cost ($) Hair Cut at least 32 Manicure at least 26 Let m = the amount of money left on the gift card.
The solution set is {m | m s gift card.
Kono knows that he can only volunteer up to 25 hours per week. If he has volunteered for the times recorded below, how much more time can Kono volunteer this week? Center Time (h) Shelter 3 h 15 min Kitchen 2 h 20 min Let t = the amount of time, in hours, left in the week for Kono to volunteer.
The solution set is . So, Kono can volunteer at most 19 hours and 25 minutes more this week.
Solve each inequality. Check your solution, and then graph it on a number line. c + (
The solution set is {c | c To check substitute three different values into the original inequality: 3.7, a number less than 3.7, and a number greater than 3.7.
The inequality is true when c is equal to or greater than 3.7, so the solution checks.
9.1g + 4.5 < 10.1g
The solution set is {g | g > 4.5}. To check substitute three different values into the original inequality: 4.5, a number less than 4.5, and a number greater than 4.5.
The inequality is true when g is greater than 4.5, so the solution checks.
The solution set is .
To check substitute three values into the original inequality: , a number less than , and a number greater than .
The inequality is true when k is greater than , so the solution checks.
The solution set is .
To check substitute three different values into the original inequality: , a number less than , and a number greater than .
The inequality is true when p is equal to or less than , so the solution checks.
In this problem, you will explore multiplication and division in inequalities.
Suppose a balance has 12 pounds on the left side and 18 pounds on the right side. Draw a picture to represent this situation. Write an inequality to represent the situation. Create a table showing the result of doubling, tripling, or quadrupling the weight on each side of the balance. Create a second table showing the result of reducing the weight on each side of the balance by a factor
of , , or . Include a column in each table for the inequality representing each situation. Describe the effect multiplying or dividing each side of an inequality by the same positive value has on the inequality. a.
12 lb < 18 lb c. 12 18 x2 24 36 x3 36 54 x4 48 72 12 18 6 9 x 4 6 x 3 x If a true inequality is multiplied by a positive number, the resulting inequality is also true. If a true inequality is divided by a positive number, the resulting inequality is also true.
If m m ?
So, ? = 17.
m + ?
So, ? = 10.
m ?
So, ? = 12.
m ?
So, ? = 3.
m ?
So, ? = 2.
m + ?
So, ? = 26.
Compare and contrast the graphs of a < 4 and a In both graphs, the line is darkened to the left of 4. In the graph of a < 4, there is a circle at 4 to indicate that 4 is not included in the graph. In the graph of a
Suppose b > d + , c + 1 < a 4, and d + a + 2. Order a, b, c, and d from least to greatest.
Since , b > d. Since , a > c. Since , d > a. So, c < a < d < b.
Write three linear inequalities that are equivalent to y < 3. To write equivalent inequalities, add or subtract the same values from both sides of the inequality. Sample answers: y + 1 < 2, y 1 < 4, y + 3 < 0.
Summarize the process of solving and graphing linear inequalities. Solving linear inequalities is similar to solving linear equations. You must isolate the variable on one side of the inequality. To graph, if the problem is a less than or a greater than inequality, an open circle is used. Otherwise a dot is used. If the variable is on the left hand side of the inequality, and the inequality sign is less than (or less than or equal to), the graph extends to the left; otherwise it extends to the right.
Explain why x 2 > 5 has the same solution set as x > 7. The inequalities are equivalent. By adding 2 to each side of the first inequality, you get the second inequality.
Which equation represents the relationship shown? x y 1 1 2 9 3 17 4 25 5 33 6 41 y = 7x 8 y = 7x + 8 y = 8x 7 y = 8x + 7 Substitute values of x and y in the equations. Choice A does not work for (1, 1).
Choice B does not work for (1, 1).
Choice D does not work for (1, 1).
Choice C works for all values.
etc. So, the correct choice is C.
What is the solution set of the inequality 7 + x < 5?
Subtracting 7 from each side of the inequality results in x < 2. So, the correct choice is H.
Francisco has $3 more than Francisco has?
Let k represent the number of dollars Kayla has. Francisco has one-fourth the number of dollars Kayla has plus three more.
The expression that represents how much money Francisco has would be . So, the correct choice is B.
The mean score for 10 students on the chemistry final exam was 178. However, the teacher had made a mistake and recorded one student s score as ten points less than the actual score. What should the mean score be? The mean is the total of the scores divided by the number of scores. So, working backwards, multiply the mean, 178, by the number of scores, 10. Then, add the 10 point mistake back to the total of the scores and divide by 10 again.
So the mean score should be 179.
Find the inverse of each function. f (x) = 7x 28
Write the slope-intercept form of an equation for the line that passes through the given point and is perpendicular to the graph of each equation. ( 2, 0), y = x 6 The slope of the line with equation y = x 6 is 1. The slope of the perpendicular line is the opposite reciprocal of 1, or 1. Find the y-intercept.
Write the equation in slope-intercept form.
( 3, 1), y = 3x + 7 The slope of the line with equation y = 3x + 7 is 3. The slope of the perpendicular line is the opposite reciprocal of
3, or . Find the y-intercept.
Write the equation in slope-intercept form.
(1, 3), y = + 4
The slope of the line with equation y = + 4 is . The slope of the perpendicular line is the opposite reciprocal of
, or 2. Find the y-intercept.
Write the equation in slope-intercept form.
( 2, 7), 2x 5y = 3 Write the equation in slope-intercept form.
The slope of the line with equation 2x 5y = 3 is . The slope of the perpendicular line is the opposite reciprocal of
, or . Find the y-intercept.
Write the equation in slope-intercept form.
On an island cruise in Hawaii, each passenger is given a lei. A crew member hands out 3 red, 3 blue, and 3 green leis in that order. If this pattern is repeated, what color lei will the 50th person receive? Let each group of 3 people with the same color lei be one term in the pattern (n). So the constant change d = 3.
th So, n needs to be in the 17 group of 3. Substitute 17 for n in the equations for each color to see which color equals 50. r = 3(17) 2 = 49 b = 3(17) 1 = 50 g = 3(17) = 51 So, the 50th person would receive a blue lei.
Find the nth term of each arithmetic sequence described.
a1 = 52, d = 12, n = 102
9, 7, 5, 3, for n = 18
0.5, 1, 1.5, 2, for n = 50
Refer to the time card shown. Write a direct variation equation relating your pay to the hours worked and find your pay if you work 30 hours.
Let p = your pay per hour.
Let y = your total pay and x = the number of hours worked. So, the direct variation equation relating your pay to the hours worked is y = 7x. Solve for x = 30.
So, if you work 30 hours, your pay is $210.
Solve each equation. 8y = 56
4p = 120
3a = 21
5-1 Solving Inequalities by Addition and Subtraction
eSolutions Manual - Powered by Cognero Page 31