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2.6
The Distributive Property
How much will you pay for six CDs?
Goal
Use the distributive
property.
When you go shopping, you can
use estimation or mental math to
determine the total cost. In Example 5
you will learn how to use the
distributive property to calculate the
total cost of six CDs—without using
a calculator.
Key Words
• distributive property
The distributive property is an important algebraic property. Example 1 uses
geometry to illustrate why the property is true for a single case. On the following
page the property is formally defined.
EXAMPLE
Student Help
VOCABULARY TIP
To distribute means to
give something to each
member of a group.
In Example 1 you can
think of the 3 as being
distributed to each
term in (x 2).
1
Use an Area Model
Find the area of a rectangle whose width is 3 and whose length is x 2.
Solution You can find the area in two ways. Remember that the area of a
rectangle is the product of the length times the width.
Area of One Rectangle
Area of Two Rectangles
3
3
x2
Area 3(x 2)
ANSWER x
2
Area 3(x) 3(2)
Because both expressions represent the same area, the following
statement is true.
Area 3(x 2) 3(x) 3(2) 3x 6
Use an Area Model
1. Write two expressions for the area of
the rectangle.
7
2. Write an algebraic statement that
shows that the two expressions
from Exercise 1 are equal.
100
Chapter 2
Properties of Real Numbers
3
x
Page 2 of 7
Student Help
STUDY TIP
Although four versions
of the distributive
property are listed, the
last three versions can
be derived from the first:
a(b c) ab ac
THE DISTRIBUTIVE PROPERTY
The product of a and (b c):
a(b c) ab ac
Example: 5(x 2) 5x 10
(b c)a ba ca
Example: (x 4)8 8x 32
The product of a and (b c):
a(b c) ab ac
Example: 4(x 7) 4x 28
(b c)a ba ca
Example: (x 5)9 9x 45
2
EXAMPLE
Use the Distributive Property with Addition
Use the distributive property to rewrite the expression without parentheses.
a. 2(x 5)
b. (1 2n)8
Solution
a. 2(x 5) 2(x) 2(5)
Distribute 2 to each term of (x 5).
2x 10
Multiply.
b. (1 2n)8 (1)8 (2n)8
8 16n
Student Help
MORE EXAMPLES
Multiply.
Use the Distributive Property with Subtraction
Use the distributive property to rewrite the expression without parentheses.
NE
ER T
More examples
are available at
www.mcdougallittell.com
INT
3
EXAMPLE
Distribute 8 to each term of (1 2n).
1
b. (2x 4)
2
a. 3(1 y)
Solution
a. 3(1 y) 3(1) 3(y)
Distribute 3 to each term of (1 y).
3 3y
Multiply.
1
1
1
b. (2x 4) (2x) (4)
2
2
2
x2
1
2
Distribute to each term of (2x 4).
Multiply.
Use the Distributive Property
Use the distributive property to rewrite the expression without
parentheses.
3. 5(n 3)
4. (2p 6)3
5. 2(x 5)
2.6
2
6. (3y 9)
3
The Distributive Property
101
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Use the Distributive Property
4
EXAMPLE
Student Help
Use the distributive property to rewrite the expression without parentheses.
STUDY TIP
Be careful when using
the distributive property
with negative factors.
Forgetting to distribute
the negative sign is a
common error.
a. 3(x 4)
b. (y 5)(4)
c. (6 3x)
d. (1 t)(9)
Solution
a. 3(x 4) 3(x) (3)(4)
Use distributive property.
3x 12
Multiply.
b. (y 5)(4) (y)(4) (5)(4)
4y 20
Multiply.
c. (6 3x) 1(6) (1)(3x)
Use distributive property.
6 3x
Multiply.
d. (1 t)(9) (1)(9) (t)(9)
9 9t
Use distributive property.
Use distributive property.
Multiply.
Use the Distributive Property
Use the distributive property to rewrite the expression without
parentheses.
7. 5(a 2)
8. (x 7)(3)
9. (4 2x)
10. (4 m)(2)
History
5
EXAMPLE
Mental Math Calculations
You are shopping for compact discs. You want to buy
6 compact discs for $11.95 each including tax. Use the distributive property
to mentally calculate the total cost of the compact discs.
COMPACT DISCS
Solution If you think of $11.95 as $12.00 $.05, the mental math is easier.
6(11.95) 6(12 0.05)
COMPACT DISCS CDs
were released in the United
States in 1984. That year
the average cost of a CD
was $17.81.
ANSWER Write 11.95 as a difference.
6(12) 6(0.05)
Use distributive property.
72 0.30
Find products mentally.
71.70
Find difference mentally.
The total cost of 6 compact discs at $11.95 each is $71.70.
Mental Math Calculations
Use the distributive property to mentally calculate the total cost.
11. You are buying birthday cards for 3 of your friends. Each card costs $1.25.
What is the total cost of the cards?
102
Chapter 2
Properties of Real Numbers
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2.6 Exercises
Guided Practice
Vocabulary Check
Explain how you would use the distributive property to simplify
the expression.
1. 2(x 3)
Skill Check
2. (x 4)5
3. 7(x 3)
4. (x 6)4
Use the area model shown.
x
5. Write two expressions for the area
of the rectangle.
5
6. Write an algebraic statement that
12
shows that the two expressions
from Exercise 5 are equal.
Match the expression with its simplified form.
7. 3(x 2)
8. (x 3)(2)
9. 3(x 2)
A. 6 2x
B. 3x 6
C. 2x 6
10. (3 x)2
D. 3x 6
Use the distributive property and mental math to simplify the expression.
( )
? (
? )
? (
? )
? ?
?
11. 4(1.15) 4(1 0.15)
12. 9(1.95) 9 ? ?
? ) ? (
? )
(
? ?
?
?
Practice and Applications
AREA MODEL Use the area model to find two expressions for the area of
the rectangle. Then write an algebraic statement that shows the two
expressions are equal.
13.
14.
2
3
x
HOMEWORK HELP
Example 1: Exs. 13–16,
71–73
Example 2: Exs. 17–28
Example 3: Exs. 29–40
Example 4: Exs. 41–54
Example 5: Exs. 55–70
x
4
Student Help
7
15.
16.
2x
11
4
x
5
9
2.6
The Distributive Property
103
Page 5 of 7
DISTRIBUTIVE PROPERTY WITH ADDITION Use the distributive property
to rewrite the expression without parentheses.
17. 3(x 4)
18. 5(w 6)
19. 7(1 t)
20. (y 4)5
21. (2 u)6
22. (x 8)7
23. 2(2y 1)
24. (3x 7)4
25. 3(4 6a)
26. (9 3n)2
27. (x 2)1.3
1
28. (10 15r)
5
DISTRIBUTIVE PROPERTY WITH SUBTRACTION Use the distributive
property to rewrite the expression without parentheses.
29. 5(y 2)
30. 2(x 3)
31. 9(7 a)
32. (x 2)2
33. (7 m)4
34. (n 7)2
35. 10(1 3t)
36. 7(6w 1)
37. (3x 3)6
38. (9 5a)4
39. (3.1u 0.8)3
1
2
40. 5 x 10
15
DISTRIBUTIVE PROPERTY Use the distributive property to rewrite the
expression without parentheses.
41. 3(r 8)
42. 2(x 6)
43. (1 s)
44. (2 t)(2)
45. (y 9)(1)
46. (x 4)(3)
47. 6(4a 3)
48. (9x 1)(7)
49. (6y 5)
50. (3d 8)(5)
51. (2.3 7w)(6)
3
52. x 24
8
ERROR ANALYSIS In Exercises 53 and 54, find and correct the error.
53.
simplify the expression.
HOMEWORK HELP
Extra help with
problem solving in
Exs. 55–66 is available at
www.mcdougallittell.com
INT
–2(7 – 8) = –2(7) – 2(8)
= –14 – 6
= –30
MENTAL MATH Use the distributive property and mental math to
Student Help
NE
ER T
54.
9(3 – 5) = 9(3) – 5
= 27 – 5
= 22
55. 4(6.11)
56. 10(7.25)
57. 3(9.20)
58. 7(5.98)
59. 2(2.90)
60. 6(8.75)
61. 3(4.10)
62. 9(1.02)
63. 2(11.05)
64. 8(2.80)
65. 5(10.99)
66. 4(5.95)
67. CONTACT LENS SUPPLIES The saline solution that you use to clean your
contact lenses is on sale for $4.99 a bottle. You decide to stock up and buy
4 bottles. Use the distributive property to mentally calculate the total cost of
the bottles of saline.
104
Chapter 2
Properties of Real Numbers
Page 6 of 7
68. SCHOOL SUPPLIES You are shopping for school supplies. You want to buy
Careers
7 notebooks for $1.05 each. Use the distributive property to mentally
calculate the total cost of the notebooks.
69. DECORATIONS You are volunteering for a charity that is operating a
haunted house. You are sent to the store to buy 5 bags of cotton balls that will
be used to make spider web decorations. Each bag costs $2.09. Use the
distributive property to mentally calculate the total cost of the cotton balls.
70. GROCERIES You see a sign at the grocery store that reads, “Buy
2 half-gallons of frozen yogurt and get one free.” Each half-gallon of frozen
yogurt costs $4.95. You decide to get three half-gallons. Use the distributive
property to mentally calculate the total cost of the frozen yogurt.
AGRONOMISTS use soil
and plant science to help
farmers increase crop
production and maintain
soil fertility.
FARMING You are trying to determine the size of a cornfield, so you will
know how many rows of corn to plant. Let x be the width of the
cornfield. Use the diagram of the farming field shown.
71. Use the diagram to find two
expressions for the area of the
entire field.
corn
x yd
soybeans
225 yd
72. Write an algebraic statement that
shows the two expressions from
Exercise 71 are equal.
73. You decide to plant the cornfield
so that x 75 yards. What is the
area of the entire field? Use one
expression from Exercise 71 to
find the solution and the other to
check your solution.
200 yd
Standardized Test
Practice
74. MULTIPLE CHOICE Which expression is equivalent to (x 7)3?
A
C
B
D
x 21
3x 10
3x 7
3x 21
75. MULTIPLE CHOICE Which expression is equivalent to 6(x 2)?
F
H
G
J
6x 2
6x 2
6x 12
6x 12
76. MULTIPLE CHOICE Which expression is equivalent to (5 x)(17)?
A
C
B
D
5 17x
85 17x
5 17x
85 17x
77. MULTIPLE CHOICE You are buying 5 new shirts to wear to school. All of
the shirts are on sale for $19.99. What expression would you use to
mentally calculate the total cost of the shirts?
F
5(20) 0.01
G
5(20 0.01)
H
2.6
5(20) 0.01
J
5(20 0.01)
The Distributive Property
105
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Mixed Review
EXPRESSIONS WITH FRACTION BARS Evaluate the expression. Then
simplify the answer. (Lesson 1.3)
10 p 8
78. 42 p 4
62 12
79. 32 1
75 52
80. 13 3 p 4
3p79
81. 24 5 11
4 p 2 53
82. 32 2
6 72
83. 3
3 97
NAMING PROPERTIES Name the property shown by the statement.
(Lesson 2.3)
84. 10 (25) 25 (10)
85. 19 0 19
86. 32 (32) 0
87. (13 8) 7 13 (8 7)
EVALUATING EXPRESSIONS Evaluate the expression. (Lesson 2.4)
Maintaining Skills
88. 6 7
89. 9 (3)
90. 4 8 3
91. 6 (8) 11
92. 7.2 9 8.5
1
2
93. 1
3
3
DECIMALS AND FRACTIONS Write the decimal as a fraction in simplest
form. (Skills Review p. 767)
94. 0.14
95. 0.25
96. 0.34
97. 0.50
98. 0.75
99. 0.82
100. 0.90
101. 0.96
Quiz 2
Evaluate the function when x 3, 1, 1, and 3. Organize your results in
a table. (Lesson 2.4)
1
1. y x 12
2. y 27 x
3. y x 4
Find the terms of the expression. (Lesson 2.4)
4. 2x 9
5. 8 x
6. 10x 4
7. STOCK MARKET The daily closing prices for a company’s stock are
$19.63, $19.88, $20.00, $19.88, and $19.75. Find the day-to-day change in
the closing price. (Lesson 2.4)
Find the product. (Lesson 2.5)
8. 7(9)
11. 1.8(6)
9. 5(6)
1
12. 15 5
10. 35(80)
13. 10(3)(9)
Rewrite the expression without parentheses. (Lesson 2.6)
14. (x 2)11
15. 5(12 y)
16. 4(3a 4)
17. SHOPPING You want to buy 2 pairs of jeans for $24.95 each. Use the
distributive property to mentally calculate the total cost of the jeans.
(Lesson 2.6)
106
Chapter 2
Properties of Real Numbers