Lesson 4.1: the Distributive Property

Chapter 4 Study Guide

Lesson 4.1: The Distributive Property

Example 1 Evaluate Numeric Expressions

Use the Distributive Property to write each expression as an equivalent expression. Then evaluate the expression. a. 3(5 – 2)

3(5 – 2) = 3  5 – 3  2 = 15 – 6 Multiply. = 9 Subtract. b. (4 + 6)2

(4 + 6)2 = 4  2 + 6  2 = 8 + 12 Multiply. = 20 Add.

Real-World Example 2 Use the Distributive Property

UNIFORMS The uniform for a fast-food restaurant job costs $39. Find the total cost of uniforms for 7 new employees.

Understand You know how many uniforms are needed and how each uniform costs. You need to find the total cost of the uniforms.

Plan You can use the Distributive Property and mental math to find the total cost of the uniforms. To find the total cost mentally, find 7($40 – $1).

Solve 7($40 – $1) = 7($40) − 7($1) Distributive Property = $280 − $7 Multiply. = $273 Subtract.

The total cost is $273.

Check You can check your result by multiplying 7  $40 or $280. The answer seems reasonable.  Example 3 Simplify Algebraic Expressions

Use the Distributive Property to write each expression as an equivalent algebraic expression. a. 4(3 + m)

4(3 + m) = 4  3 + 4m = 12 + 4m Simplify. b. (p + 2)5

(p + 2)5 = p  5 + 2  5 = 5p + 10 Simplify.

Example 4 Simplify Expressions with Subtraction

Use the Distributive Property to write each expression as an equivalent algebraic expression.

** If you have a subtraction sign, change it to addition and change the second number to its opposite. ** a. 5(w – 7) 5(w – 7) = 5[w + (-7)] Rewrite w – 7 as w + (-7). = 5  w + 5  (-7) Distributive Property = 5w + (-35) Simplify. = 5w – 35 Definition of subtraction b. –2(y – 4) -2(y – 4) = -2[y + (-4)] Rewrite y – 4 as y + (-4). = -2  y + (-2)(-4) Distributive Property = -2y + 8 Simplify. Lesson 4.1 ½ : Factoring Linear Expressions

A monomial is a number, a variable, or a product of a number and one or more variabl es.

Monomials Not Monomials 25, x, 40x x + 4, 40x+120

To factor a number means to write it as a product of its factors. A monomial can be factored using the same method that you would use to factor a number. The greatest common factor (GCF) of two monomials is the greatest monomial that is a factor of both.

Example 1: Find the GCF of Monomials

a.) Find the GCF of 4x and 12x

Think, what are the factors?: 4x = 1, 2, 4, x Write the factors of 4x and 12x

12x = 1, 2 , 3, 4, 6, 12, x

4x and 12x have factors of 1, 4, x

The GCF of 4x and 12x is 4x.

b.) Find the GCF of 18a and 20ab

Think, what are the factors?: 18a = 1, 2, 9, 18, a Write the factors of 18a and 20ab

20ab = 1, 2, 4, 5, 20, a, b

18a and 20ab have 1, 2, a in common.

The GCF of 18a and 20ab is 2a. You use the Distributive Property and the work backward strategy to express an algebraic expression as a product of its factors. An algebraic expression is in factored form when it is expressed as the product of its factors.

8x + 4 = 4(2x) + 4(1)

4(2x + 1)

Example 2: Factor Algebraic Expressions

a.) 3x + 9

Think, what are the factors?: Find the GCF: 3x = 1, 3, x 3x and 9 have 1, 3 in common.

9 = 1, 3, 9

The GCF of 3x and 9 is 3.

3x + 9 = 3(x) + 3(3)

3(x+3)

b.) 12x + 7

Think, what are the factors?: Find the GCF: 12x= 1, 2, 3, 4, 6, 12, x 12x and 7 have 1 in common.

7 = 1, 7

Since the only common factor is 1, 12x +7 cannot be factored.

c.) 4x + 28

Think, what are the factors?: Find the GCF: 4x = 1, 2, 4, x 4x and 28 have 1, 2, 4 in common.

28 = 1, 2, 4, 7, 14

The GCF of 4x and 28 is 4. 4x + 28 = 4(x) + 4(7)

4(x + 7)

Lesson 4.2: Simplifying Expressions

Example 1 Identify Like Terms

Identify the like terms in the following expressions.

a. 7y + 1 + 8y 7y and 8y are like terms since the variables are the same.

b. 9a + 2 + 6 + 4a 9a and 4a are like terms since the variables are the same. Constant terms 2 and 6 are also like terms.

Example 2 Identify Parts of an Expression

Identify the terms, like terms, coefficients, and constants in the expression

x + 5y – 2y – 3

x + 5y – 2y – 3 = x + 5y + (-2y) + (-3) Definition of subtraction = 1x + 5y + (-2y) + (-3) Identity Property

The terms are x, 5y, -2y, and -3. The like terms are 5y and -2y. The coefficients are 1, 5, and -2. The constant is -3. Example 3 Simplify Algebraic Expressions

Simplify each expression. a. 3x + 9 + 4x

3x + 9 + 4x = 3x + 4x + 9 Commutative Property = (3 + 4)x + 9 Distributive Property = 7x + 9 Simplify. b. 8 – 5m + m – 3

8 – 5m + m – 3 = 8 + (-5m) + m + (-3) Definition of subtraction = 8 + (-5m) + (1m) + (-3) Identity Property = (-5m) + (1m) + 8 + (-3) Commutative Property = (-5 + 1)m + 8 + (-3) Distributive Property = -4m + 5 Simplify. c. a + 5(2a + 3b)

a + 5(2a + 3b) = a + 5(2a) + 5(3b) Distributive Property = a + 10a + 15b Simplify. = 1a + 10a + 15b Identity Property = (1 + 10)a + 15b Distributive Property = 11a + 15b Simplify.

Real-World Example 4 Write and Simplify Algebraic Expressions

AGES Emily and Kate are sisters. Emily is four years younger than Kate. Write an expression in simplest form that represents the sum of Emily and Kate’s ages.

Words Emily’s age plus Kate’s age

Variables Let x = Kate’s age. Let x – 4 = Emily’s age.

Expression (x – 4) + x

(x – 4) + x = x + (x – 4) Commutative Property = (x + x) – 4 Associative Property = (1x + 1x) – 4 Identity Property = (1 + 1)x – 4 Distributive Property = 2x – 4 Simplify.

The expression 2x – 4 represents the sum of Emily and Kate’s ages.

Lesson 4.3: Solving Equations b y Addition or Subtraction

Example 1 Solve Equations by Adding

Solve each equation. Check your solution and graph it on a number line. a. –3 + x = 18

–3 + x = 18 Write the equation. + 3 = + 3 Addition Property of Equality x = 21

To check that 21 is the solution, replace x with 21 in the original equation.

Check –3 + x = 18 Write the equation.

–3 + 21 ≟ 18 Check to see whether this sentence is true.

18 = 18  The sentence is true.

The solution is 21. To graph 21, draw a dot at 21 on a number line.

–513 –4 14 –3 15 –216 –117 180 191 202 213 224 235

b. 8.2 = q – 1.5

8.2 = q – 1.5 Write the equation.

8.2= q + -1.5 Change the subtraction to “add the opposite” + 1.5 = + 1.5 Addition Property of Equality 9.7 = q

The solution is 9.7. Check your solution. To graph 9.7, draw a dot at 9.7 on a number line. 9.7  1 2 3 4 5 6 7 8 9 10 12 Example 2 Solving Equations by Subtracting

Solve each equation. Check your solution. a. 9 + b = 4

9 + b = 4 Write the equation. – 9 = – 9 Subtraction Property of Equality b = –5 Additive Inverse and Identity Properties

To check that –5 is the solution, replace b with –5 in the original equation.

Check 9 + b = 4 Write the equation. ? 9 + (-5)  4 Check to see whether this sentence is true. 4 = 4  The sentence is true.

b. 2.8 + y = 0.4

2.8 + y = 0.4 Write the equation. – 2.8 = – 2.8 Subtraction Property of Equality y = –2.4 Additive Inverse and Identity Properties

The solution is -2.4. Check your solution.

Real-World Example 3 Solve by Subtracting

BASEBALL Carlos had 12 more hits during baseball season than Peter. Carlos had 73 hits during the season. Write and solve an equation to find the number of hits that Peter had.

Words Carlos’s hits equals Peter’s hits + 12.

Variable Let x = the number of hits that Peter had.

Equation 73 = x + 12 73 = x + 12 Write the equation. – 12= – 12 Subtraction Property of Equality. 61 = x Additive Inverse and Identity Properties

Peter had 61 hits.

Lesson 4.4: Solving Equations by Multiplying or Dividing Lesson 4.5: Solving Two Step Equations Example 3 Solve a Two-Step Equation y Solve 8 + = 14. Check your solution. 3 y 8 + = 14 Write the equation. 3 -8 = – 8 Subtraction Property of Equality y = 6 Simplify. 3 y 3  = 3  6 Multiplication Property of Equality 3 y = 18 Simplify. Check your solution.

Example 4 Equations with Negative Coefficients

Solve 6 = 9 – x. Check your solution. 6 = 9 – x Write the equation. 6 = 9 – 1x Identity Property; x = 1x 6 = 9 + (-1x) Definition of subtraction -9 + 6 = -9 + 9 + (-1x) Addition Property of Equality -3 = -1x Simplify.  3 1x = Division Property of Equality 1 1 3 = x Simplify. Check your solution.

Example 5 Combine Like Terms Before Solving

Solve 3w – 7w – 11 = 25. Check your solution. 3w – 7w – 11 = 25 Write the equation. -4w – 11 = 25 Distributive Property; 3w – 7w = –4w -4w +-11 = 25 Change subtraction to “add the opposite” +11 = +11 Addition Property of Equality -4w = 36 Simplify.  4w 36 = Division Property of Equality  4  4 w = -9 Simplify. Check your solution. Lesson 4.6: Writing Equations

Example 2 Translate Sentences into Equations

Translate each sentence into an equation.

Sentence Equation a. Three less than twice a number is –12. 2n – 3 = -12 b. Mara has 15 ponytail holders, which is 6 more than 3 times the number that Janelle has. 15 = 3n + 6 c. The quotient of a number and 4, decreased by 2, is 5. - 2 = 5

Example 3 Write and Solve an Equation

BASKETBALL Olivia scored twenty-four points in a basketball game. This is six more than three times the number of points that Madison scored. Find the number of points that Madison scored.

Let n = the number of points that Madison scored. Then, 3n + 6 = the number of points that Olivia scored.

24 = 3n + 6 Write the equation. – 6 = – 6 Subtraction Property of Equality 18 = 3n Simplify. 6 = n Mentally divide each side by 3.

Madison scored 6 points.

Real-World Example 4 Solve a Two-Step Verbal Problem