P.2
Exercises
VOCABULARY CHECK: Fill in the blanks.
1. The three components of a ________ are a set of numbers, operations with the set of numbers,
and properties of the numbers and operations.
2. The basic properties of a mathematical system are often called ________.
3. The formal argument that justifies a theorem is called a ________.
In Exercises 1–28, name the property of real numbers that
justifies the statement.
25. 36 b 3
63b
26. x 1 x 1 0
1. 3 5 5 3
2. 57 75
27. 32 x 3
3. 25 25 0
4. 5 0 5
28. 6 x m 6 x m
5. 610 106
6. 26
7. 7 1 7
8. 4
3 2 63
1
41
36y 8
30. Commutative Property of Addition
11. 3 12 9 3 12 9
10 6 12. 16 8 5 16 8 5
10 5 10
79 15 7 9 7 15
31. Commutative Property of Multiplication
13. 8 510 8
14.
153 32. Associative Property of Addition
15. 10 8 3 10 8 3
6 5 y 16. 5 108 85 10
17. 52a 5
2a
18. 102x 10
2x
19. 1 5t 5t
20. 8y 1 8y
21. 3x 0 3x
22. 0 8w 8w
23.
1
y
y1
In Exercises 29–38, use the property of real numbers to fill
in the missing part of the statement.
29. Associative Property of Multiplication
9. 25 35 35 25
10. 4 10 8 410
2 3x
24. 10x 1
1
10x
33. Distributive Property
6 z5 34. Distributive Property
34 x 35. Commutative Property of Addition
25 x Section P.2
36. Additive Inverse Property
Properties of Real Numbers
In Exercises 71–74, identify the property of real numbers
that justifies each step.
13x 13x x53
37. Multiplicative Identity Property
71.
x 8 1 38. Additive Identity Property
8x 0 x 5 5 3 5
x 5 5 2
x 0 2
x 2
72.
x 8 20
x 8 8 20 8
x 8 8 28
x 0 28
x 28
73.
2x 5 6
2x 5 5 6 5
2x 5 5 11
2x 0 11
2x 11
1
1
2 2x 2 11
12 2x 112
1 x 11
2
x 11
2
74.
3x 4 10
3x 4 4 10 4
3x 4 4 6
3x 0 6
3x 6
1
1
3 3x 3 6
13 3x 2
1x2
x2
In Exercises 39–46, give (a) the additive inverse and (b) the
multiplicative inverse of the quantity.
39. 10
40. 18
41. 16
42. 52
43. 6z, z 0
44. 2y, y 0
45. x 1, x 1
46. y 4, y 4
In Exercises 47–54, rewrite the expression using the
Associative Property of Addition or the Associative Property
of Multiplication.
47. x 5 3
48. z 6 10
49. 32 4 y
50. 15 3 x
51. 34
5
53. 62y
52. 10
8 5
54. 83x
In Exercises 55–62, rewrite the expression using the
Distributive Property.
55. 202 5
56. 34 8
57. 53x 4
58. 62x 5
59. x 62
60. z 1012
61. 62y 5
62. 410 b
In Exercises 63–68, the right side of the equation is not
equal to the left side. Change the right side so that it is
equal to the left side.
63. 3x 5 3x 5
64. 4x 2 4x 2
65. 2x 8 2x 16
66. 9x 4 9x 36
67. 303 1
68. 6
1
6
0
In Exercises 69 and 70, use the properties of real
numbers to prove the statement.
69. If ac bc and c 0, then a b.
70. 1a a
23
Original equation
Original equation
Original equation
Original equation
In Exercises 75–80, use the Distributive Property to perform
the arithmetic mentally. For example, you work in an industry where the wage is $14 per hour with “time and a half ”
for overtime. So, your hourly wage for overtime is
141.5 14 1 1
2
14 7
$21.
75. 161.75 162 14 76. 15123 152 13 77. 762 760 2
78. 549 550 1
24
Chapter P
Prerequisites
79. 96.98 97 0.02
86. Geometry The figure shows two adjoining rectangles. Find the total area of the two rectangles in two
ways.
80. 1219.95 1220 0.05
x
Number of Warehouses
In Exercises 81–84, the number
of Costco warehouses for the years 1997 through 2004 are
approximated by the expression
8
6
23.4t 89.
In this expression, t represents the year, with t 7 corresponding to 1997 (see figure). (Source: Costco
Wholesale)
Synthesis
Number of warehouses
y
True or False?
In Exercises 87–90, determine whether
the statement is true or false. Justify your answer.
450
400
350
87. 6x 6x 0
300
88. 9 5 5 9
250
89. 67 2 67 2
t
7
8
9
10
11
12
13
14
Year (7 ↔ 1997)
81. Use the graph to approximate the number of
warehouses in 2000.
82. Use the expression to approximate the annual
increase in the number of warehouses.
83. Use the expression to predict the number of
warehouses in 2007.
84. In 2003, the actual number of warehouses was 397.
Compare this with the approximation given by the
expression.
85. Geometry The figure shows two adjoining rectangles. Find the total area of the rectangles in two
ways.
x
3
90. 48 1 48 41
91. Think About It Does every real number have a
multiplicative inverse? Explain.
92. What is the additive inverse of a real number? Give
an example of the Additive Inverse Property.
93. What is the multiplicative inverse of a real number?
Give an example of the Multiplicative Inverse
Property.
94. State the Multiplication Property of Zero.
95. Writing Explain how the Addition Property of
Equality can be used to allow you to subtract the
same number from each side of an equation.
96. Investigation You define a new mathematical
operation using the symbol . This operation is
defined as a b 2 a b.
(a) Is this operation commutative? Explain.
(b) Is this operation associative? Explain.
4
50
(1-50)
1. 7
Chapter 1
Real Numbers and Their Properties
EXERCISES
Reading and Writing After reading this section write out the
answers to these questions. Use complete sentences.
1. What is the difference between the commutative property
of addition and the associative property of addition?
The commutative property says that a b b a and
the associative property says that (a b) c a (b c).
2. Which property involves two different operations?
The distributive property involves multiplication and
addition.
3. What is factoring?
Factoring is the process of writing an expression or number
as a product.
4. Which two numbers play a prominent role in the properties
studied here?
The number 0 is the additive identity and the number 1 is
the multiplicative identity.
5. What is the purpose of studying the properties of real
numbers?
The properties help us to understand the operations and
how they are related to each other.
6. What is the relationship between rate and time?
If one task is completed in x hours, then the rate is 1x tasks
per hour.
Use the commutative property of addition to rewrite each
expression. See Example 1.
7. 9 r
8. t 6
9. 3(2 x)
r9
6t
3(x 2)
10. P(1 rt)
11. 4 5x
12. b 2a
P(rt 1)
5x 4
2a b
Use the commutative property of multiplication to rewrite each
expression. See Example 2.
13. x 6
14. y (9)
15. (x 4)(2)
6x
9y
2(x 4)
16. a(b c)
(b c)a
17. 4 y 8
4 8y
18. z 9 2
9z 2
Use the commutative and associative properties of multiplication and exponential notation to rewrite each product. See
Example 3.
19. (4w)(w)
20. (y)(2y)
21. 3a(ba)
4w 2
2y 2
3a 2b
22. (x x)(7x)
23. (x)(9x)(xz)
24. y( y 5)(wy)
7x 3
9x 3z
5y 3w
Evaluate by finding first the sum of the positive numbers and
then the sum of the negative numbers. See Example 4.
25.
26.
27.
28.
8 4 3 10 3
3 5 12 10 0
8 10 7 8 7 10
6 11 7 9 13 2 4
29.
30.
31.
32.
4 11 7 8 15 20 21
8 13 9 15 7 22 5 29
3.2 2.4 2.8 5.8 1.6 0.6
5.4 5.1 6.6 2.3 9.1 13.7
33. 3.26 13.41 5.1 12.35 5
22.4
34. 5.89 6.1 8.58 6.06 2.34 0.03
Use the distributive property to rewrite each product as a sum
or difference and each sum or difference as a product. See
Example 5.
35. 3(x 5) 3x 15
36. 4(b 1) 4b 4
37. 2m 12 2(m 6)
38. 3y 6 3(y 2)
39. a(2 t) 2a at
40. b(a w) ab bw
41. 3(w 6) 3w 18 42. 3(m 5) 3m 15
43. 4(5 y) 20 4y 44. 3(6 p) 18 3p
45. 4x 4 4(x 1)
46. 6y 6 6(y 1)
47. 1(a 7) a 7
48. 1(c 8) c 8
49. 1(t 4) t 4
50. 1(x 7) x 7
51. 4y 16 4(y 4)
52. 5x 15 5(x 3)
53. 4a 8 4(a 2)
54. 7a 35 7(a 5)
Find the multiplicative inverse (reciprocal) of each number. See
Example 6.
1
1
1
55. 2
56. 3
57. 5 2
3
5
1
1
1
58. 6 59. 7 60. 8 6
7
8
61. 1 1
62. 1 1
63. 0.25 4
4
2
2
64. 0.75 65. 2.5 66. 3.5 3
5
7
Name the property that justifies each equation. See Example 7.
67. 3 x x 3 Commutative property of multiplication
68. x 5 5 x Commutative property of addition
69. 2(x 3) 2x 6 Distributive property
70. a(bc) (ab)c Associative property of multiplication
71. 3(xy) (3x)y Associative property of multiplication
72. 3(x 1) 3x 3 Distributive property
73. 4 (4) 0 Inverse properties
74. 1.3 9 9 1.3 Commutative property of addition
75. x 2 5 5x 2 Commutative property of multiplication
76. 0 0 Multiplication property of 0
77. 1 3y 3y Identity property
78. (0.1)(10) 1 Inverse property
79. 2a 5a (2 5)a Distributive property
80. 3 0 3 Identity property
81. 7 7 0 Inverse property
82. 1 b b Identity property
83. (2346)0 0 Multiplication property of 0
1.7
Distributive property
85. ay y y(a 1)
Distributive property
86. ab bc b(a c) Distributive property
Complete each equation, using the property named.
87. a y ____, commutative
ya
88. 6x 6 ____, distributive
89. 5(aw) ____, associative
6(x 1)
(5a)w
90. x 3 ____, commutative
1
1
91. x ____, distributive
2
2
3x
1
(x
2
92. 3(x 7) ____, distributive
93. 6x 15 ____, distributive
104. Farmland conversion. The amount of farmland in the
United States is decreasing by one acre every 0.00876
hours as farmland is being converted to nonfarm use
(American Farmland Trust, www.farmland.org). At what
rate in acres per day is the farmland decreasing?
2740 acres/day
1)
3x 21
3(2x 5)
94. (x 6) 1 ____, associative
x (6 1)
95. 4(0.25) ____, inverse property
1
96. 1(5 y) ____, distributive
5 y
97. 0 96(____), multiplication property of zero
98. 3 (____) 3, identity property
1
99. 0.33(____) 1, inverse property
100
33
100. 8(1) ____, identity property
0
8
Solve each problem. See Example 8.
101. Laying bricks. A bricklayer lays one brick in 0.04 hour,
while his apprentice lays one brick in 0.05 hour.
a) If both are working, then at what combined rate (in
bricks per hour) are they laying bricks?
45 bricks/hour
b) Which person is working faster?
Bricklayer
Number of bricks laid
200
150
Bricklayer
100
Apprentice
50
0
0
2
4
6
Time (hours)
(1-51) 51
103. Population explosion. In 1998, the population of the
earth was increasing by one person every 0.3801 second
(World Population Data Sheet 1998, www.prb.org).
a) At what rate in people per second is the population of
the earth increasing?
2.63 people/second
b) At what rate in people per week is the population of
the earth increasing?
1,591,160 people/week
8
FIGURE FOR EXERCISE 101
102. Recovering golf balls. Susan and Joan are diving for golf
balls in a large water trap. Susan recovers a golf ball every
0.016 hour while Joan recovers a ball every 0.025 hour. If
both are working, then at what rate (in golf balls per hour)
are they recovering golf balls?
102.5 balls/hour
Farmland (millions of acres)
84. 4x 4 4(x 1)
Properties of the Real Numbers
1000
950
900
1990
2000
Year
2010
FIGURE FOR EXERCISE 104
GET TING MORE INVOLVED
105. Writing. The perimeter of a rectangle is the sum of twice
the length and twice the width. Write in words another
way to find the perimeter that illustrates the distributive
property.
The perimeter is twice the sum of the length and width.
106. Discussion. Eldrid bought a loaf of bread for $1.69 and
a gallon of milk for $2.29. Using a tax rate of 5%, he correctly figured that the tax on the bread would be 8 cents
and the tax on the milk would be 11 cents, for a total of
$4.17. However, at the cash register he was correctly
charged $4.18. How could this happen? Which property
of the real numbers is in question in this case?
Due to rounding off, the tax on each item seperately does
not equal the tax on the total. It looks like the distributive
property fails.
107. Exploration. Determine whether each of the following
pairs of tasks are “commutative.” That is, does the order in
which they are performed produce the same result?
a) Put on your coat; put on your hat.
Commutative
b) Put on your shirt; put on your coat.
Not commutative
Find another pair of “commutative” tasks and another pair
of “noncommutative” tasks.