Algebra I Handout #4 (Self Study Guide) 2012

Algebra I Handout #4 (Self Study Guide) 2012 This handout is the summary of Algebra I you learned so far. (We did not go over Key #3 and Key #9 yet) Do NOT lose this handout! You must study this handout to pass Algebra I final exam and to pass CAHSEE (California High School Exit Exam) Math Part. Key #1. Remember the meaning of “Opposite” and “Reciprocal”. Opposite  Opposite Sign (Just change the sign +  -, -  +) Reciprocal  You flip the number (upside down)  the Sign stays the same!  Sometimes, we call “Reciprocal” as “Multiplicative Inverse” 4 4 Examples: Opposite of 5 = -5, Opposite of -3 = 3, Opposite of   5 5 3 7 x ay 2 y 2x Reciprocal of = , Reciprocal of = , Reciprocal of  =  , 7 3 ay 2 x 2x y 1 5 1 Reciprocal of 5 = (remember 5  ), Reciprocal of  = -5 5 1 5 Key #2. Exponent is the most important in CAHSEE – Math. x7 x2 x3  x23  x5 (Add exponents)  x72  x5 (Subtract Exponents) x 2

More Examples: x5 x2  x7 , x8 x  x9 (Remember x  x1 and x0 1 )

7 5 5 x x x 3 1  x2 ,  x4 ,  x7 (* 5 – (-2) = 7) Negative Exponent  x  x5 x x2 x3

Which expression is equivalent to x2 x7 ? A) x8 x4 B) x7 x4 C) x6 x4 D) x5 x4

26  24 2x2 1 1 1  2645  25 ,   x3   , 7a 2b 8bc2  56a 2b2c2 ,(3a 2b)2  3a 2b  3a 2b 25 10x5 5 5 x3

Key #3. Square Root (2 9  9  32  3) and Cubic Root (3 8  3 23  2) Recall, Square Root and Square are cancelled, Cubic Root and Cube are cancelled.

Example: 22  4, 32  9, 42  16, 52  25, 62  36, 7 2  49, cube 23  8, 33  27, 43  64, 53  125 4  2, 9  3, 16  4, 25  5, 36  6, 49  7, 3 8  2, 3 27  3, 3 64  4

36  4  6  2  4 , 3 25  2 16  35  2 4 15  8  23 , 36x2  6x , 25x4  5x2

49  3 8  7  2  5, 3 64  3 81  4  39  4  27  31, 3 125  64  5  8 13

KWON MHS 1/5 Page Algebra I Handout #4 (Self Study Guide) 2012 Key #4. Integer Operations (-) + (-) is always (-)  (-3) + (-5) = -8, (-12) + (-13) = -25, Negative + Negative = (-) (-) x (-) is always (+)  (-3) x (-5) = 15, (-7) x (-8) = 56, Negative x Negative = Positive (+) x (-) = (-) Positive x Negative = Negative (-) x (+) = (-) Negative x Positive = Negative 3 – (-2) = 3 + 2 (neg neg  +) -3 + 10 = +7, 5 – 9 = -4, -6 + -3 = -9, -5 – 4 = -5 + -4 = -9 7 – 17 = -10, -3 – 9 = -3 + -9 = -12, -6 + 10 = 4, -5 + -3 = -8 -4 x -3 = 12, -5 x 2 = -10, 7 x -3 = -21, 8 x 7 = 56 Example: Which of the following expressions results in a negative number? a) -3 + 7 b) -4 + -3 c) 3 + (-7) + 5 d) 7 – 5 e) 2 – (-5) Key #5. Fractions (Add, Subtract, Multiply, Divide) 2 1 CROSS PRODUCT 4 2  31 11 3 2 35  4  2 23     Ex:    3 4 Multiply Bottoms 3 4 12 4 5 45 20 3 2 CROSS PRODUCT 33  4 2 1 1 1 13  2 1 1     Ex:    4 3 Multiply Bottoms 43 12 2 3 23 6 2 5 MultiplyTop totop 25 10 3 3 33 9     Ex:    3 7 Multiply Bottomtobottom 3 7 21 5 4 45 20 3 2 3 2 3 3 33 9      & Flip the second fraction      4 3 4 3 4 2 4 2 8 If the bottom is the same, then keep the bottom! 11 7 11 7 4 1 7 1 7 1 8 4     ,     12 12 12 12 3 10 10 10 10 5 Key #6. Absolute Value | -5 | = 5, | 0 | = 0, | -20 | = 20, | 17 | = 17 What is the absolute value of -4? |-4| = 4 Which number has the greatest absolute value? A) 17 B) -13 C) 15 D) -20 | x | = 3. What are possible values for x? {-3, 3} | x + 2 | = 9. What are the possible values for x? x+2 = 9 or x+2 = -9  x=7 or x=-11 | 2x – 2 | = 4. What are the possible values for x? 2x-2=4 or 2x-2=-4  x=__ or x=__

Key #7. Line y = mx + b; m = slope, b = Y-intercept y = -3x + 5. Slope = -3, Y-Intercept = 5. Two lines are PARALLEL. Then the two lines should have the same SLOPE. If the slope is positive, then line goes up. Negative slope  falls down. Slope = Rise over Run. Y-intercept  Set X be 0. X-intercept  Set Y be 0. 1 A line parallel to the line y=4x-7 is A)y  x  7 B) y  4x  3 C) y  4x  3 4

KWON MHS 2/5 Page Algebra I Handout #4 (Self Study Guide) 2012

Key #8. How to Graph a Line (y = mx + b) Step 1: Change any form to y=mx+b (4x + 2y = 8  y= -2x + 4) Step 2: Start with y-intercept. y = -2x + 4  y-intercept is 4.  From origin(0,0), go up 4 step. Let’s call this point as “Starting point” Step 3: Look at the slope (-2 here) and use “Rise over Run”. -2 over 1  Go down 2 and Run 1 step. Let’s call this point as “Ending point”. Step 4: Connect “Starting point” (y-intercept) and “Ending point”.  2 Graph y = 2x + 3 Graph y = -x - 1 Graph y  x  2 3

Find Y-intercept and Slope from below Graph.

Key #9. Solve or Graph of Linear Inequality (y > mx + b or y < mx + b) Inequality has no equality symbol (“=”). There are 4 inequality symbols. (,,,) “>”  “ greater than ”, ""  “greater than or equal to”, “<”  “less than ”, ""  “less than or equal to”. Graph: “>, <” Dotted line ,", "Solid line

Example: 2x + 4 > 6 (Solve as you solved 2x+4=6)  2x +4 -4 > 6 -4  2x > 2  x>1 Solve -2x -4 < 6  Add 4 both sides  -2x -4 +4 < 6 + 4  -2x < 10  We have to divide by 2 for both sides. Whenever we multiply or divide by negative number in inequality, the symbols must be reversed ***  -2x < 10  -2x /(-2) > 10/(-2)  here the symbol “ < ” is changed by “ > ” because we divided by negative number (-2)  The final answer is x > -5.

KWON MHS 3/5 Page Algebra I Handout #4 (Self Study Guide) 2012 Graph y  2x  3 y  2x  3 y  2x  3 y  2x  3

Key #10. Line: Finding Y-intercept and Finding X-intercept. y = -3x + 6. Y-intercept is 6 by setting x=0. (Ignore x when you find Y-intercept) y = -3x + 6. To find X-intercept, set y=0. 0=-3x+6. x should be 2. X-intercept is 2. Remember! Y-interceptx=0, X-intercepty=0. Example 1 : 2x – 3y = 12. Find y-intercept and x-intercept. y-intercept  x=0 (ignore x part)  -3y=12  y=-4.(y-intercept) x-intercept  y=0 (ignore y part)  2x = 12  x=6 (x-intercept) Example 2: -3x + 4y = 24. Find x-intercept and y-intercept.

Key #1 1 . Solving Equations and “Distributive Property” * Solve equation (=Find a solution): 2x + 3 = 9  Leave x alone  subtract 3 from both sides  2x + 3 – 3 = 9 – 3  2x = 6  Divide by 2 both sides  x = 3 * 2(x+2) = 6. Use “Distributive Property”! Whenever you see parenthesis() in equations, use “Distribute”.  2x + 4 = 6  2x=2  x=1 * Solve 5x – 2(x-2) = -5  Find a solution. x = ____ 2x 2x 2x * Equations with fraction.  2  4   2  2  4  2   6  2x  63 x=9 3 3 3

Key #12. Remember! Whenever you see (#, #) form, that is a POINT (x, y). For example, (1, 2)  point x=1 and y=2. (0, 2)  point x=0 and y=2. You need to use “Plug-in & Plug-in” method to solve the problems. Example: Which point lies on the line of 3x + 2y = 2. A.(0,2) B.(0,1) C.(0,1) D.(2,0) Let’s plug-in for (0,2)  (x,y)  x=0, y=2 plug-in 3x+2y=2. 3(0)+2(2)=2? NO! Let’s plug-in for (0,1)  (x,y)  x=0, y=1 plug-in 3x+2y=2. 3(0)+2(1)=2? YES! This is it! Example: Which pair is the solution to the system of equation below? x + 2y = 3, 2x – y = 1 A.(-1,2) B.(1,1) C.(0,1) D.(1,-1) Let’s plug-in for (-1,2)  x=-1,y=2 plug-in x+2y=3  -1+2(2)=3? YES!  x=-1,y=2 plug-in 2x-y=1  2(-1)-(2)=1? NO! This is not it. Let’s plug-in for (1,1)  x=1,y=1 plug-in x+2y=3  1+2(1)=3? YES!  x=1,y=1 plug-in 2x-y=1  2(1)-1=1? YES! Both YES! This is it!

KWON MHS 4/5 Page Algebra I Handout #4 (Self Study Guide) 2012

Key #13. Integer Operations

(-) + (-) is always (-)  (-3) + (-5) = -8, (-12) + (-13) = -25, Negative + Negative = (-) (-) x (-) is always (+)  (-3) x (-5) = 15, (-7) x (-8) = 56, Negative x Negative = Positive (+) x (-) = (-) Positive x Negative = Negative (-) x (+) = (-) Negative x Positive = Negative 3 – (-2) = 3 + 2 (neg neg  +) -3 + 10 = +7, 5 – 9 = -4, -6 + -3 = -9, -5 – 4 = -5 + -4 = -9 7 – 17 = -7, -3 – 9 = -3 + -9 = -12, -6 + 10 = 4, -5 + -3 = -8 -4 x -3 = 12, -5 x 2 = -10, 7 x -3 = -21, 8 x 7 = 56

KWON MHS 5/5 Page