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Prealgebra

7.3 Distributive Property

Prealgebra7.3 Distributive Property
  1. Preface
  2. 1 Whole Numbers
    1. Introduction
    2. 1.1 Introduction to Whole Numbers
    3. 1.2 Add Whole Numbers
    4. 1.3 Subtract Whole Numbers
    5. 1.4 Multiply Whole Numbers
    6. 1.5 Divide Whole Numbers
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  3. 2 The Language of Algebra
    1. Introduction to the Language of Algebra
    2. 2.1 Use the Language of Algebra
    3. 2.2 Evaluate, Simplify, and Translate Expressions
    4. 2.3 Solving Equations Using the Subtraction and Addition Properties of Equality
    5. 2.4 Find Multiples and Factors
    6. 2.5 Prime Factorization and the Least Common Multiple
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  4. 3 Integers
    1. Introduction to Integers
    2. 3.1 Introduction to Integers
    3. 3.2 Add Integers
    4. 3.3 Subtract Integers
    5. 3.4 Multiply and Divide Integers
    6. 3.5 Solve Equations Using Integers; The Division Property of Equality
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  5. 4 Fractions
    1. Introduction to Fractions
    2. 4.1 Visualize Fractions
    3. 4.2 Multiply and Divide Fractions
    4. 4.3 Multiply and Divide Mixed Numbers and Complex Fractions
    5. 4.4 Add and Subtract Fractions with Common Denominators
    6. 4.5 Add and Subtract Fractions with Different Denominators
    7. 4.6 Add and Subtract Mixed Numbers
    8. 4.7 Solve Equations with Fractions
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  6. 5 Decimals
    1. Introduction to Decimals
    2. 5.1 Decimals
    3. 5.2 Decimal Operations
    4. 5.3 Decimals and Fractions
    5. 5.4 Solve Equations with Decimals
    6. 5.5 Averages and Probability
    7. 5.6 Ratios and Rate
    8. 5.7 Simplify and Use Square Roots
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  7. 6 Percents
    1. Introduction to Percents
    2. 6.1 Understand Percent
    3. 6.2 Solve General Applications of Percent
    4. 6.3 Solve Sales Tax, Commission, and Discount Applications
    5. 6.4 Solve Simple Interest Applications
    6. 6.5 Solve Proportions and their Applications
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  8. 7 The Properties of Real Numbers
    1. Introduction to the Properties of Real Numbers
    2. 7.1 Rational and Irrational Numbers
    3. 7.2 Commutative and Associative Properties
    4. 7.3 Distributive Property
    5. 7.4 Properties of Identity, Inverses, and Zero
    6. 7.5 Systems of Measurement
    7. Key Terms
    8. Key Concepts
    9. Exercises
      1. Review Exercises
      2. Practice Test
  9. 8 Solving Linear Equations
    1. Introduction to Solving Linear Equations
    2. 8.1 Solve Equations Using the Subtraction and Addition Properties of Equality
    3. 8.2 Solve Equations Using the Division and Multiplication Properties of Equality
    4. 8.3 Solve Equations with Variables and Constants on Both Sides
    5. 8.4 Solve Equations with Fraction or Decimal Coefficients
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  10. 9 Math Models and Geometry
    1. Introduction
    2. 9.1 Use a Problem Solving Strategy
    3. 9.2 Solve Money Applications
    4. 9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem
    5. 9.4 Use Properties of Rectangles, Triangles, and Trapezoids
    6. 9.5 Solve Geometry Applications: Circles and Irregular Figures
    7. 9.6 Solve Geometry Applications: Volume and Surface Area
    8. 9.7 Solve a Formula for a Specific Variable
    9. Key Terms
    10. Key Concepts
    11. Exercises
      1. Review Exercises
      2. Practice Test
  11. 10 Polynomials
    1. Introduction to Polynomials
    2. 10.1 Add and Subtract Polynomials
    3. 10.2 Use Multiplication Properties of Exponents
    4. 10.3 Multiply Polynomials
    5. 10.4 Divide Monomials
    6. 10.5 Integer Exponents and Scientific Notation
    7. 10.6 Introduction to Factoring Polynomials
    8. Key Terms
    9. Key Concepts
    10. Exercises
      1. Review Exercises
      2. Practice Test
  12. 11 Graphs
    1. Graphs
    2. 11.1 Use the Rectangular Coordinate System
    3. 11.2 Graphing Linear Equations
    4. 11.3 Graphing with Intercepts
    5. 11.4 Understand Slope of a Line
    6. Key Terms
    7. Key Concepts
    8. Exercises
      1. Review Exercises
      2. Practice Test
  13. A | Cumulative Review
  14. B | Powers and Roots Tables
  15. C | Geometric Formulas
  16. Answer Key
    1. Chapter 1
    2. Chapter 2
    3. Chapter 3
    4. Chapter 4
    5. Chapter 5
    6. Chapter 6
    7. Chapter 7
    8. Chapter 8
    9. Chapter 9
    10. Chapter 10
    11. Chapter 11
  17. Index

Learning Objectives

By the end of this section, you will be able to:
  • Simplify expressions using the distributive property
  • Evaluate expressions using the distributive property
Be Prepared 7.3

Before you get started, take this readiness quiz.

  1. Multiply: 3(0.25).3(0.25).
    If you missed this problem, review Example 5.15
  2. Simplify: 10(−2)(3).10(−2)(3).
    If you missed this problem, review Example 3.51
  3. Combine like terms: 9y+17+3y2.9y+17+3y2.
    If you missed this problem, review Example 2.22.

Simplify Expressions Using the Distributive Property

Suppose three friends are going to the movies. They each need $9.25;$9.25; that is, 99 dollars and 11 quarter. How much money do they need all together? You can think about the dollars separately from the quarters.

The image shows the equation 3 times 9 equal to 27. Below the 3 is an image of three people. Below the 9 is an image of 9 one dollar bills. Below the 27 is an image of three groups of 9 one dollar bills for a total of 27 one dollar bills. The image shows the equation 3 times 25 cents equal to 75 cents. Below the 3 is an image of three people. Below the 25 cents is an image of a quarter. Below the 75 cents is an image of three quarters.

They need 33 times $9,$9, so $27,$27, and 33 times 11 quarter, so 7575 cents. In total, they need $27.75.$27.75.

If you think about doing the math in this way, you are using the Distributive Property.

Distributive Property

If a,b,ca,b,c are real numbers, then

a(b+c)=ab+aca(b+c)=ab+ac

Back to our friends at the movies, we could show the math steps we take to find the total amount of money they need like this:

3(9.25) 3(9+0.25) 3(9)+3(0.25) 27+0.75 27.75 3(9.25) 3(9+0.25) 3(9)+3(0.25) 27+0.75 27.75

In algebra, we use the Distributive Property to remove parentheses as we simplify expressions. For example, if we are asked to simplify the expression 3(x+4),3(x+4), the order of operations says to work in the parentheses first. But we cannot add xx and 4,4, since they are not like terms. So we use the Distributive Property, as shown in Example 7.19.

Example 7.17

Simplify: 3(x+4).3(x+4).

Try It 7.33

Simplify: 4(x+2).4(x+2).

Try It 7.34

Simplify: 6(x+7).6(x+7).

Some students find it helpful to draw in arrows to remind them how to use the Distributive Property. Then the first step in Example 7.17 would look like this:

The image shows the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4. The image shows and equation. On the left side of the equation is the expression x plus 4 in parentheses with the number 3 outside the parentheses on the left. There are two arrows pointing from the top of the three. One arrow points to the top of the x. The other arrow points to the top of the 4. This is set equal to 3 times x plus 3 times 4.

Example 7.18

Simplify: 6(5y+1).6(5y+1).

Try It 7.35

Simplify: 9(3y+8).9(3y+8).

Try It 7.36

Simplify: 5(5w+9).5(5w+9).

The distributive property can be used to simplify expressions that look slightly different from a(b+c).a(b+c). Here are two other forms.

Distributive Property

If a,b,ca,b,c are real numbers, then

a(b+c)=ab+aca(b+c)=ab+ac

Other forms

a(bc)=abaca(bc)=abac
(b+c)a=ba+ca(b+c)a=ba+ca

Example 7.19

Simplify: 2(x3).2(x3).

Try It 7.37

Simplify: 7(x6).7(x6).

Try It 7.38

Simplify: 8(x5).8(x5).

Do you remember how to multiply a fraction by a whole number? We’ll need to do that in the next two examples.

Example 7.20

Simplify: 34(n+12).34(n+12).

Try It 7.39

Simplify: 25(p+10).25(p+10).

Try It 7.40

Simplify: 37(u+21).37(u+21).

Example 7.21

Simplify: 8(38x+14).8(38x+14).

Try It 7.41

Simplify: 6(56y+12).6(56y+12).

Try It 7.42

Simplify: 12(13n+34).12(13n+34).

Using the Distributive Property as shown in the next example will be very useful when we solve money applications later.

Example 7.22

Simplify: 100(0.3+0.25q).100(0.3+0.25q).

Try It 7.43

Simplify: 100(0.7+0.15p).100(0.7+0.15p).

Try It 7.44

Simplify: 100(0.04+0.35d).100(0.04+0.35d).

In the next example we’ll multiply by a variable. We’ll need to do this in a later chapter.

Example 7.23

Simplify: m(n4).m(n4).

Try It 7.45

Simplify: r(s2).r(s2).

Try It 7.46

Simplify: y(z8).y(z8).

The next example will use the ‘backwards’ form of the Distributive Property, (b+c)a=ba+ca.(b+c)a=ba+ca.

Example 7.24

Simplify: (x+8)p.(x+8)p.

Try It 7.47

Simplify: (x+2)p.(x+2)p.

Try It 7.48

Simplify: (y+4)q.(y+4)q.

When you distribute a negative number, you need to be extra careful to get the signs correct.

Example 7.25

Simplify: −2(4y+1).−2(4y+1).

Try It 7.49

Simplify: −3(6m+5).−3(6m+5).

Try It 7.50

Simplify: −6(8n+11).−6(8n+11).

Example 7.26

Simplify: −11(43a).−11(43a).

Try It 7.51

Simplify: −5(23a).−5(23a).

Try It 7.52

Simplify: −7(815y).−7(815y).

In the next example, we will show how to use the Distributive Property to find the opposite of an expression. Remember, a=−1·a.a=−1·a.

Example 7.27

Simplify: (y+5).(y+5).

Try It 7.53

Simplify: (z11).(z11).

Try It 7.54

Simplify: (x4).(x4).

Sometimes we need to use the Distributive Property as part of the order of operations. Start by looking at the parentheses. If the expression inside the parentheses cannot be simplified, the next step would be multiply using the distributive property, which removes the parentheses. The next two examples will illustrate this.

Example 7.28

Simplify: 82(x+3).82(x+3).

Try It 7.55

Simplify: 93(x+2).93(x+2).

Try It 7.56

Simplify: 7x5(x+4).7x5(x+4).

Example 7.29

Simplify: 4(x8)(x+3).4(x8)(x+3).

Try It 7.57

Simplify: 6(x9)(x+12).6(x9)(x+12).

Try It 7.58

Simplify: 8(x1)(x+5).8(x1)(x+5).

Evaluate Expressions Using the Distributive Property

Some students need to be convinced that the Distributive Property always works.

In the examples below, we will practice evaluating some of the expressions from previous examples; in part , we will evaluate the form with parentheses, and in part we will evaluate the form we got after distributing. If we evaluate both expressions correctly, this will show that they are indeed equal.

Example 7.30

When y=10y=10 evaluate: 6(5y+1)6(5y+1) 6·5y+6·1.6·5y+6·1.

Try It 7.59

Evaluate when w=3:w=3:5(5w+9)5(5w+9)5·5w+5·9.5·5w+5·9.

Try It 7.60

Evaluate when y=2:y=2:9(3y+8)9(3y+8)9·3y+9·8.9·3y+9·8.

Example 7.31

When y=3,y=3, evaluate −2(4y+1)−2(4y+1)−2·4y+(−2)·1.−2·4y+(−2)·1.

Try It 7.61

Evaluate when n=−2:n=−2:−6(8n+11)−6(8n+11)−6·8n+(−6)·11.−6·8n+(−6)·11.

Try It 7.62

Evaluate when m=−1:m=−1:−3(6m+5)−3(6m+5)−3·6m+(−3)·5.−3·6m+(−3)·5.

Example 7.32

When y=35y=35 evaluate (y+5)(y+5) and y5y5 to show that (y+5)=y5.(y+5)=y5.

Try It 7.63

Evaluate when x=36:x=36:(x4)(x4)x+4x+4 to show that (x4)=x+4.(x4)=x+4.

Try It 7.64

Evaluate when z=55:z=55:(z10)(z10)z+10z+10 to show that (z10)=z+10.(z10)=z+10.

Media Access Additional Online Resources

Section 7.3 Exercises

Practice Makes Perfect

Simplify Expressions Using the Distributive Property

In the following exercises, simplify using the distributive property.

90.

4(x+8)4(x+8)

91.

3(a+9)3(a+9)

92.

8(4y+9)8(4y+9)

93.

9(3w+7)9(3w+7)

94.

6(c13)6(c13)

95.

7(y13)7(y13)

96.

7(3p8)7(3p8)

97.

5(7u4)5(7u4)

98.

12(n+8)12(n+8)

99.

13(u+9)13(u+9)

100.

14(3q+12)14(3q+12)

101.

15(4m+20)15(4m+20)

102.

9(59y13)9(59y13)

103.

10(310x25)10(310x25)

104.

12(14+23r)12(14+23r)

105.

12(16+34s)12(16+34s)

106.

r(s18)r(s18)

107.

u(v10)u(v10)

108.

(y+4)p(y+4)p

109.

(a+7)x(a+7)x

110.

−2(y+13)−2(y+13)

111.

−3(a+11)−3(a+11)

112.

−7(4p+1)−7(4p+1)

113.

−9(9a+4)−9(9a+4)

114.

−3(x6)−3(x6)

115.

−4(q7)−4(q7)

116.

−9(3a7)−9(3a7)

117.

−6(7x8)−6(7x8)

118.

(r+7)(r+7)

119.

(q+11)(q+11)

120.

(3x7)(3x7)

121.

(5p4)(5p4)

122.

5+9(n6)5+9(n6)

123.

12+8(u1)12+8(u1)

124.

163(y+8)163(y+8)

125.

184(x+2)184(x+2)

126.

411(3c2)411(3c2)

127.

96(7n5)96(7n5)

128.

22(a+3)22(a+3)

129.

8(r7)8(r7)

130.

−12(u+10)−12(u+10)

131.

−4(c10)−4(c10)

132.

(5m3)(m+7)(5m3)(m+7)

133.

(4y1)(y2)(4y1)(y2)

134.

5(2n+9)+12(n3)5(2n+9)+12(n3)

135.

9(5u+8)+2(u6)9(5u+8)+2(u6)

136.

9(8x3)(−2)9(8x3)(−2)

137.

4(6x1)(−8)4(6x1)(−8)

138.

14(c1)8(c6)14(c1)8(c6)

139.

11(n7)5(n1)11(n7)5(n1)

140.

6(7y+8)(30y15)6(7y+8)(30y15)

141.

7(3n+9)(4n13)7(3n+9)(4n13)

Evaluate Expressions Using the Distributive Property

In the following exercises, evaluate both expressions for the given value.

142.

If v=−2,v=−2, evaluate

  1. 6(4v+7)6(4v+7)
  2. 6·4v+6·76·4v+6·7
143.

If u=−1,u=−1, evaluate

  1. 8(5u+12)8(5u+12)
  2. 8·5u+8·128·5u+8·12
144.

If n=23,n=23, evaluate

  1. 3(n+56)3(n+56)
  2. 3·n+3·563·n+3·56
145.

If y=34,y=34, evaluate

  1. 4(y+38)4(y+38)
  2. 4·y+4·384·y+4·38
146.

If y=712,y=712, evaluate

  1. −3(4y+15)−3(4y+15)
  2. 3·4y+(−3)·153·4y+(−3)·15
147.

If p=2330,p=2330, evaluate

  1. −6(5p+11)−6(5p+11)
  2. −6·5p+(−6)·11−6·5p+(−6)·11
148.

If m=0.4,m=0.4, evaluate

  1. −10(3m0.9)−10(3m0.9)
  2. −10·3m(−10)(0.9)−10·3m(−10)(0.9)
149.

If n=0.75,n=0.75, evaluate

  1. −100(5n+1.5)−100(5n+1.5)
  2. −100·5n+(−100)(1.5)−100·5n+(−100)(1.5)
150.

If y=−25,y=−25, evaluate

  1. (y25)(y25)
  2. y+25y+25
151.

If w=−80,w=−80, evaluate

  1. (w80)(w80)
  2. w+80w+80
152.

If p=0.19,p=0.19, evaluate

  1. (p+0.72)(p+0.72)
  2. p0.72p0.72
153.

If q=0.55,q=0.55, evaluate

  1. (q+0.48)(q+0.48)
  2. q0.48q0.48

Everyday Math

154.

Buying by the case Joe can buy his favorite ice tea at a convenience store for $1.99$1.99 per bottle. At the grocery store, he can buy a case of 1212 bottles for $23.88.$23.88.

  1. Use the distributive property to find the cost of 1212 bottles bought individually at the convenience store. (Hint: notice that $1.99$1.99 is $2$0.01.$2$0.01.)

  2. Is it a bargain to buy the iced tea at the grocery store by the case?

155.

Multi-pack purchase Adele’s shampoo sells for $3.97$3.97 per bottle at the drug store. At the warehouse store, the same shampoo is sold as a 3-pack3-pack for $10.49.$10.49.

  1. Show how you can use the distributive property to find the cost of 33 bottles bought individually at the drug store.

  2. How much would Adele save by buying the 3-pack3-pack at the warehouse store?

Writing Exercises

156.

Simplify 8(x14)8(x14) using the distributive property and explain each step.

157.

Explain how you can multiply 4($5.97)4($5.97) without paper or a calculator by thinking of $5.97$5.97 as 60.0360.03 and then using the distributive property.

Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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