Chapter 1 Variables, Real Numbers, and Mathematical Models
54. 00
8
55. 7
0 is undefined.
61.
1
180 30 180 6
30
62.
1
150 25 150 6
25
63.
040
64.
0100
65. 40 is undefined.
66. 10 0 is undefined.
71. 14 7 14 8
98 97
112 7 16 16
63 7 9 9
76. 2972
88 36
922
77.
52 52 10
xxx
78.
93 93 27
xxx
79. 33
443
44
yyy
80. 33
553
55
yyy
Section 1.7 Multiplication and Division of Real Numbers
89.
4234243812
xx x
94.
31 31
yy
95.
42 3 7 2
yy
42 4 3 7 2
8127 2
87122
14
yy
yy
yy
y
97. 4210
4( 5) 2( 5) 10
20 10 10
20 20, true
xx
The number is a solution.
100. 421715
4( 2) 21 7( 2) 15
yy
35 41, false
The number is not a solution.
102. 6( 2) 4 10
6( 9 2) 4( 9) 10
6( 7) 36 10
42 46, false
ww
The number is not a solution.
00,true
The number is a solution.
104.
5(7 ) 12 0
5 7 ( 5) 12( 5) 0
57 5 60 0
zz
Chapter 1 Variables, Real Numbers, and Mathematical Models
106. 16 4 2 21
xx
107. 5132
64
5( 4) 1 3( 4) 2
64
20 1 12 2
64
21 14
64
77
,true
22
mm
The number is a solution.
109.
410 8 408 32
110.
315 14 4514 31
111.
93 227229
20
118. 5( 32)
9
55
(31 32) (63) 35
99
CF
C
31 F is equivalent to 35 C.
119. a. 2012 is 47 years after 1965.
0.5 42
0.5(47) 42
18.5
Cx
C
According to the formula, about 18.5% of
0.5 42
0.5(40) 42
22
Cx
C
According to the formula, about 22% of
American adults smoked in 2005.
This overestimates the actual amount shown in
the bar graph by 1%.
Section 1.7 Multiplication and Division of Real Numbers
b. 0.7 41
0.7(10) 41
34
Ln
According to the formula, 34% of Americans in
2007 preferred larger families.
This is the same as the estimate in part a.
122. a. According to the graph, about 56% of
Americans in 2007 preferred smaller families.
b.
0.6 51
0.6(10) 51
57
Ln
the graphs.
123. – 130. Answers will vary.
131. does not make sense; Explanations will vary.
Sample explanation: The sign rules for dividing real
numbers and multiplying real numbers are the same.
132. makes sense
135. false; Changes to make the statement true will vary.
A sample change is: The sum of two negative
numbers is a negative number.
136. false; Changes to make the statement true will vary.
A sample change is: If the number is multiplied by
0, which is nonnegative, the result is 0.
141.
12
x
142.
40
40x
143. The calculator verifies your results.
148.
63633
149.
1
636 2
3
150.
2
(6) (6)(6) 36
Chapter 1 Variables, Real Numbers, and Mathematical Models
1.8 Check Points
1. a.
2
66636
2. a.
22 2 2
16 5 (16 5) 21xx x x
b.
33 3 3 3 3
771(71)8xx x x x x
c.
23
10 8xx
cannot be simplified.
3.
20 4 3 17 20 12 17
20 12 17
15
6.
22 22
178 11
( 18) ( 18)
21015 26
11 18
8.
33
25 5 3[4 2(7 9) ] 25 5 3[4 2( 2) ]
25 5 3[4 2( 8)]
25 5 3[4 16]
81
21
25 30
7
5
7
10.
22
4 ( 5) 4( 5)
25 20
5
xx
According to the model, males between the ages of
19 and 30 with this lifestyle need 2662 calories per
day. This underestimates the actual value shown in
the bar graph by 38 calories.
Section 1.8 Exponents and Order or Operations
c.
30 300,000
30(100,000) 300,000 3,000,000 300,000
100, 000 100,000
3, 300, 000
100,000
33
x
Cx
C
The average cost is $33.
1.8 Concept and Vocabulary Check
1. base; exponent
2. b to the nth power
1.8 Exercise Set
1.
2
99981
10.
6
1 1111(1)(1)1
11.
4
5 5555 625
12.
6
1 111111 1
13.
2
10 10 10 100
14.
2
88864
15.
22 2 2
712 712 19xx x x
16.
22 2 2
618 618 24xx x x
17.
33 3 3
10 5 10 5 15xx x x
18.
33 3 3
14 8 14 8 22xx x x
22
1
xx
22.
22 2 2
2
29 30 29 30
29 30
xx x x
x
Chapter 1 Variables, Real Numbers, and Mathematical Models
25.
23
55xx
cannot be simplified. The
27.
22 2 2
2
2
16 16 16 16
16 16
00
xx x x
x
x
28.
22 2 2
2
2
34 34 1
34 1
33
xx x x
x
x
36.
36 12 4 2 36 3 2
33 2 35
37.
22
8 16 2 4 3 64 16 4 4 3
39.
22
32 43 3449
45 32
45 32
13
41.
222
45 45 20 425
400 100
300
42.
222
35 35 15 325
45.
33
33
63 5 21 3
62 22
68 28
Section 1.8 Exponents and Order or Operations
49.
25 29 4 25 25
51.
3
732 1 21 7381 21
737 21
721 21
28 21
28 7 4
21 7 3
4
3
53.
22
10 8 18 18 2
25 16 9
54
54.
22
6 4 36 16 20 2
2 ( 8) 2 8 10
57 2 34 7 5(5) 3 3
59. 4 10 (8 20) 4 10 ( 12) 4 10 12
422 4 22
88
60. 6 7 4 3 6 7 12 6 5 6 5 30
61.
810 45 80 20
80 20 60
62.
4 15 3 10 60 30
9220 4
92 5 910 19
65.
2
39
24 6 24 6
85 3
Chapter 1 Variables, Real Numbers, and Mathematical Models
10 5
69. 91 3 5 91 36
42 4 6 42 45
918
820
45 36 9
40 40 40
71.
7727
333 33
999
55
24 24
66
20
33
9
524
6
25
17 52
5
17 5 5 1
25 17 1 2
425 1
425 2
11
122
75.
2
38; 2
xxx
2
2
383292
3 4 8 2 12 16 28
xx
76.
2
42; 3xxx
2
2
424323
49 2 3 36 6 42
xx
Section 1.8 Exponents and Order or Operations
79.
2
2
64
; 5
yy
y
80.
2
32
; 5
yy
y
5
33
3
81.
35 2 1 35 10 1
35 9
15 27
xx
x
x
82.
46 3 1 46 18 1
46 17
24 68
xx
x
x
85.
743 4 5
y
7434 5
y
86.
658 2 4
y
22
22
61081
xx
2
24 12 10 1
14 11
xx
x
89.
3
10 2 10 8 10 8 2
90.
3
100 5 100 125
100 125
25
22
2
96.
33
46424xx
Chapter 1 Variables, Real Numbers, and Mathematical Models
98.
2
96 802 660
Mx x
99. a.
2
2
2 170 5115
5
2 10 170 10 5115
5
1323
nn
M
c.
1323 985 338
The difference is $338. This is $6 less than the
difference shown by the graph.
100. a.
2
2 170 5115
5
nn
M
2
3 145 3775
nn
According to the formula, the median weekly
earning of a female college graduate in 2005 is
$885.
This overestimates the earnings shown on the
bar graph by $2.
c.
1183 885 298
c. The cost of cleanup increases as the percentage
of contaminant removed increases.
102. a.
200
100
200(60) 300
x
Cx
103. – 105. Answers will vary.
106. makes sense
Chapter 1 Review Exercises
107. does not make sense; Explanations will vary.
Sample explanation:
4
10 10,000
111. false; Changes to make the statement true will vary.
A sample change is:
6( 3) 6 18 6 12 6
31 2 2
and
6(2) 6 12 6 18 6
21 3 3
114.
111
62 8
439
111
610
439
115. (2 3 3) 5 (6 3) 5 9 5 45
116.
1
25 10 9 10 5 9 59 45
2
119. Answers will vary. One example is 5.
13
26
11
,true
22
The number is a solution.
44 2 3( 17) 8
42 51 8
42 43, false
The number is not a solution.
50
3.
40 40 10
585
200 80
40 40
y
x
Chapter 1 Variables, Real Numbers, and Mathematical Models
4. 3(2 ) 3(2(10) 8)
3(20 8)
3(28)
84
yx
9. 4513
4(3) 5 13
12 5 13
17 13, false
x
The number is not a solution.
11. 3( 1) 11 2( 8)
3(2 1) 11 2(2 8)
3(3) 11 2(10)
911 20
20 20, true
ww
The number is a solution.
12. According to the line graph, the average number of
Latin words that the class remembered after 5 days
was about 11.
14. a. 594 21,348
594(13) 21,348
29,070
Cn
According to the formula, the average annual
15. 237221223
377 77
16. 9511955964
511 11 11 11
55
19. Composite
60 6 10 2 3 2 5 2 2 3 5
20. Composite
63 79 733 337
21. 67 is a prime number.
22. 15 3 5 5
33 3 11 11
Chapter 1 Review Exercises
29. 32 31524458 37
4 15 4 15 15 4 60 60 60
30. 37
x
31. 213
3155
213
22
3155
ww
32. 11
224
xx
33. 3(6)
5
x
37.
0.625
8 5.000
48
38.
11 3.0000…
22
80
77
11
39. a.
81 9
b.
0, 81
c.
17,0, 81
Chapter 1 Variables, Real Numbers, and Mathematical Models
41. Answers will vary. One example of a rational
number that is not an integer is
3
4
.
3
46.
111
; 0.25 is to the left of
454
111
0.2, so .
545
51. 713 13 7yy
52.
97 79xx
53.
64 64 10yyy
54.
7 10 7 10 70xxx
57.
63 4 52 1xx
63 64 52 5 1
18 24 10 5
xx
xx
59.
8113
60.
31 3514
62.
8 6 12 11xy xy
812 611
812 611
45 or 54
xxyy
xy
xy yx
63.
10(3 4) ( 40 ) 30 40 ( 40 )
30 ( 40 ) 40
yyy y
yy
Chapter 1 Review Exercises
68.
71 715
10 2 10 2 5
75 12 6
10 10 10 5
71.
25 4 10 16
25 4 10 16
25 4 10 16
29 26
3
74.
71284
75.
35 35 3
511 511 11
76.
53 2 4 120
81.
32 1 4 5
xx
32 3 1 4 5
6 345
xx
xx
14 2, false
The number is not a solution.
83.
2( 3) 18 5
2( 4 3) 18 5( 4)
2( 1) 18 20
218 20
xx
2014
2
B
For the years 2000 and 2014 the mathematical
model gives the exact number shown by the graph.
85.
2
66636
86.
2
66636
Chapter 1 Variables, Real Numbers, and Mathematical Models
92.
65 32 64 1 6410
95.
8453 8415
811 88
96.
6103 67
215 93 3027
42 14
3
99.
2
23; 1
xxx
2
2
23 1 213
123
6
xx
100.
2
7; 2
xxx
62
62 6
12 6 or 6 12
y
y
yy
103. a.
2
2
24 756 16,739
24 10 756 10 16,739
26, 699
Dn n
According to the formula, the average student
2.
911 36
20 9 11
3.
317 51
4.
315 37
77 715
21 21
105
1
21
1
5
5
Chapter 1 Test
5.
13107
31
3434
10 7 70
34 12
235
26
35 5
or 5
66
9.
23 23
68 57 2 2
48 32
10.
32 22 64
28 3 25
64 10 1
25 10
12.
53 4 2
xy xy
53 54 2
15 20 2
15 2 20
13 19
xyxy
xyxy
xx yy
xy
14. Rational numbers can be written as the quotient of
two integers.
74 4 0 1
7 , ,0 ,0.25 ,
155 1 4
22222
42 ,and .
177
Thus,
4
7, , 0, 0.25, 4 ,
5
and
22
7
are the
rational numbers of the set.
100 50 150
19.
2323
xx
20.
64 64 24
xxx
21.
75 1 2 75 71 72
35 7 14
xy x y
xy
113
(3 2) 3
5105
133
(5)
5105
36
110 10
9
1,false
Chapter 1 Variables, Real Numbers, and Mathematical Models
25.
1532
4
x
26. 5( 4) 7x
29.
4(220 )
5
44
(220 40) (180) 144
Ha
H