Chapter 1
Variables, Real Numbers, and Mathematical Models
1.1 Check Points
1. a.
62 62(10) 26x
3. a.
6x
b.
4x
x
4. a.
9342
9(6) 3 42
x
5. a.
5
6
x
b.
72 1x
6. a.
45
4(15) 5 65
dn
d
65% of marriages end in divorce after 15 years
when the wife is under 18 at the time of marriage.
1.1 Concept and Vocabulary Check
1. variable
1.1 Exercise Set
4.
16 16 4 12x
5.
55420x
11. 2( 5) 2(4 5) 2(9) 18x
12. 5( 3) 5(4 3) 5(7) 35x
13.
12 8 12 4 8 48 8 40 5
22488
x
x
14.
5 52 5452 2052 72 6
3341212
x
x
Chapter 1 Variables, Real Numbers, and Mathematical Models
18. 3( )3(75)3(12)36xy
19. 4 3 4 7 3 5 28 15 13xy
22.
50 14 50 14 10 2 8
57yx
25.
4x
26.
6x
33.
9x
34.
3x
35.
35x
36.
53x
42.
30 4
x
44.
17 22
517 22
22 22, true
x
The number is a solution.
46.
50 20
50 30 20
y
40 30, false
The number is not a solution.
49.
8
6
48 8
6
r
Section 1.1 Introduction to Algebra: Variables and Mathematical Models
51.
4323
m
52.
3419
3(6) 4 19
18 4 19
22 19, false
m
The number is not a solution.
54.
5326
5(3) 3 2(3) 6
15 3 6 6
12 12, true
aa
The number is a solution.
56.
4( 3) 6
4(6 3) 6(6)
4(9) 36
36 36, true
pp
The number is a solution.
58.
3( 2) 4( 3)
3(10 2) 4(10 3)
3(12) 4(7)
ww
59.
428x
62.
1
84
x
63.
20 5x
64.
40 10x
68.
4329x
69.
4533x
70.
6333x
71. 4( 5) 33x
72
7(5) 2 37
xy
x
Evaluate the expression.
37 5 32 8
444
xy
333
Chapter 1 Variables, Real Numbers, and Mathematical Models
77. First find x.
1
y
x
78. First find x.
1
3
y
x
79. a. 2( 3 ) 2(4 3 1) 2(7) 14xy
b.
53040
z
80. a. 3(2 ) 3(2 1 5) 3(7) 21xy
81. a. 6 2 6 3 2 6 18 12 6xy
b.
7452
7(6) 45 2(6)
ww
82. a.
1
514 5314 1578
2
xy
0.16(10) 2.83
4.43
T
According to the formula, the average price in
1990 was $4.43.
The bar graph shows a price in 2010 of $7.89.
$7.89 $7.63 $0.26
The model underestimates the actual amount
shown in the bar graph by $0.26.
1985 was $3.63.
The model overestimates the actual amount
2005 was $6.83.
The bar graph shows a price in 2005 of $6.41.
$6.83 $6.41 $0.42
The model overestimates the actual amount
shown in the bar graph by $0.42.
Section 1.2 Fractions in Algebra
86.
52 0.3
pa
87. a.
0.8(200 )
0.8(200 145)
44
HA
H
A bowler with an average score of 145 will have
a handicap of 44.
b. The bowler’s final score will be
120 44,
or
164.
89. – 97. Answers will vary.
98.
0.8(200 )
0.8(200 200) 0
HA
H
A bowler with an average score of 200 will have no
handicap.
99. makes sense
105. true
106. false; Changes to make the statement true will vary.
A sample change is: Hard work combined with a
willingness to use available learning resources can
107. Choices of variables may vary.
Let
t
number of televisions assembled.
10
tw
109.
32 32 6
75 75 35
110.
27 25 2510
35 37 37 21
95954
2. 5 divided by 3 is 1 with a remainder of 2, so
52
1
33
.
3. Begin by selecting any two numbers whose product
is 36.
Here is one possibility:
36 4 9
Because the factors 4 and 9 are not prime, factor
each of these composite numbers.
necessary to write prime factorizations.
We can use the greatest common factor to
reduce this fraction.
42 7 6
24
46
7
4
Chapter 1 Variables, Real Numbers, and Mathematical Models
c.
13
15
; Because 13 and 15 share no common
5. a.
42 42 8
11 3 11 3 33
b.
36318 3
63
515 5 5
6. a.
5358 5
48 43 4
4
210 1
3
333
b.
223212
3
331339
c.
31279189 1
31 2
888884 4
8.
22714
33721
b.
43443316 9 7
3 4 3 4 4 3 12 12 12
10.
10 2 5
12 2 2 3
LCD 2 2 3 5 60
3 7 3 6 7 5 18 35 53
7
11,true
The given fraction is a solution.
b.
11
53
ww
12. a.
2(6)
3
x
b.
31
2
45
xx
Section 1.2 Fractions in Algebra
1.2 Concept and Vocabulary Check
1. numerator; denominator
2. mixed; improper
7.
a
b
8.
ac
bd
9. reciprocals
10. d
1.2 Exercise Set
1.
328316319
288 88
7. 23 divided by 5 is 4 with a remainder of 3, so
23 3
4
55
.
8. 47 divided by 8 is 5 with a remainder of 7, so
47 7
5
88
.
99
11. 711 divided by 20 is 35 with a remainder of 11, so
711 11
35
20 20
.
12. 788 divided by 25 is 31 with a remainder of 13, so
788 13
31
25 25
.
17. 37 has no factors other than 1 and 37, so 37 is
prime.
18. 23 has no factors other than 1 and 23, so 23 is
prime.
Chapter 1 Variables, Real Numbers, and Mathematical Models
24. 83 has no factors other than 1 and 83, so 83 is
prime.
222235
28.
composite; 360 10 36
2566
252323
222335
29.
10 2 5 5
16 2 8 8
30.
8244
14 2 7 7
50 5 10 10
34.
45 5 9 9
50 5 10 10
35.
32 16 2 2
80 16 5 5
39.
120 2 60 60
86 2 43 43
43.
8 11 8 11 88
44.
53 53 15
8 11 8 11 88
45.
4949436 1
9 or 5
71717 7 7
46.
3838324 3
8 or 3
71717 7 7
47.
15 15 5 51 1
50.
76 76 42 2
4 11 4 11 44
21
2
21
22
22
51.
3 3 15 8 120 20
31
454520
6
20
6
1
52.
411457010
21
7
Section 1.2 Fractions in Algebra
55.
18 18 1
2
552
18 1 18 2 9 9 4
or 1
52 10 2 5 5 5
58.
12 3 7 21 3 7 7 3
3or1
711212344 4
59.
31 34 34123
44 41 41 4
62.
73787856414
4 8 4 3 43 12 43
14 2
or 4
33
63.
11 17 7 711
14 7 14 1 14 7 2 2
64.
11 14 4 41 1
84818422
68.
52527
13 13 13 13
69.
71 8422
12 12 12 4 3 3
72.
33 6 23 3
88 8 24 4
73.
75 2 2
12 12 12
1
2
1
6
6
76.
17 2 15 3
555
77.
1 1 15 12
2 5 25 52
52527
10 10 10 10
78.
11 1513
35 3553
53538
Chapter 1 Variables, Real Numbers, and Mathematical Models
81.
35 3352
812 83122
91019
24 24 24
83.
1121122114 7
18 9 18 9 2 18 18 18
84.
174174217 8 9
1891892181818
9
1
9
1
2
2
87.
737835
10 16 10 8 16 5
56 15 41
80 80 80
90.
21115
32
3232
11 2 5 3
32 23
7828
2
78 28
21
7
2
42
28
1
28 28, true
The given number is a solution.
30 30, true
The given number is a solution.
93.
23
34
w
Section 1.2 Fractions in Algebra
94.
37
44
w
137
224 4
537
24 4
10 3 7
444
77
,true
44
96.
11
12 42
11
12 20 20
42
120 120
12 41 21
12 5 10
710,false
zz
The given number is not a solution.
97.
21 3
937
yy
227 127 3
98.
252
36
yy
21 51
112
33 63
24 54 2
33 63
8202
918
8102
86,false
The given number is not a solution.
100.
13
(2)3 (34)
28
13
(4 2) 3 (3 4 4)
28
13
(2) 3 (12 4)
28
3
13 (8)
8
43,false
xx
The given number is not a solution.
Chapter 1 Variables, Real Numbers, and Mathematical Models
101.
27
(6) (2)
39
yy
1217
46 42
102.
15
(6) (2)
39
yy
2125
26 22
3339
103.
1
5
x
104.
1
6
x
105.
1
xx
108.
11
32
xx
1112
114.
392
4
xx
115.
333
aaa
15
68
18 8 2
65 30
4
2
4
15
15
120.
11 11 21 32
24 23 44 66
Section 1.2 Fractions in Algebra
121.
111
2
525
xx
122.
12 3 2 4 3xxx
111
12 3 3 2 4 3 3 3
222
123. 5( 32)
9
55
(68 32) (36) 20
99
CF
C
68 F is equivalent to 20 C.
125. a.
7220
10
Ha
160
The upper limit of the heart rate for a 20-year-
old with this exercise goal is 160 beats per
minute.
b.
5
3220 30
5
3190
5
114
Ha
H
Chapter 1 Variables, Real Numbers, and Mathematical Models
127. a.
9220
10
Ha
128. a.
7220
8
Ha
b.
7220
8
Ha
129. a. 2010 is 10 years after 2000.
31
100 2
31
10
100 2
Ix
I
c. 2016 is 16 years after 2000.
31
Ix
50 of adults used the
internet in 2016.
130. a. 2014 is 14 years after 2000.
31
23
25
Ix
The formula estimates that 23
25 of adults used the
internet in 2014.
Section 1.3 The Real Numbers
c. 2015 is 15 years after 2000.
31
Ix
131. – 140. Answers will vary.
141. makes sense
142. does not make sense; Explanations will vary.
Sample explanation: Fractions are often used in
algebra.
145. false; Changes to make the statement true will vary.
A sample change is: 11 5 2 7
.
2 5 10 10 10
146. false; Changes to make the statement true will vary.
A sample change is: 114111
4
221248.
149.
151. 15
2or
22
b.
c.
3. a.
24
60
56
40
40
0
30.375
60
55
50
Chapter 1 Variables, Real Numbers, and Mathematical Models
5. a. 9
b.
0, 9
6. a.
14 5 since 14 is to the right of 5 on the number
line.
b.
5.4 2.3 since 5.4 is to the left of 2.3 on the
number line.
7. a. 23 is true because 23 is true.
b.
22 is true because 22 is true.
c.
41 is false because neither 41 nor
41 is true.
1.3 Concept and Vocabulary Check
1. natural
2. whole
6. rational; irrational
7. left
4. 12,500
5. −3000
6. −3
10. 5 is shown as a dot on the number line.
11. −5 is shown as a dot on the number line.
14. 1
24 is shown as a dot on the number line.
15. 11
3 is shown as a dot on the number line.
Section 1.3 The Real Numbers
17. −1.8 is shown as a dot on the number line.
20. 11
5
is shown as a dot on the number line.
21.
0.75
43.00
22.
0.6
53.0
30
0
30.6
5
23.
0.35
20 7.00
60
25.
0.875
8 7.000
70.875
8
26.
0.3125
16 5.0000
48
20
50.3125
16
27.
0.818…
11 9.000…
88
20
11
90
88
Chapter 1 Variables, Real Numbers, and Mathematical Models
29.
0.5
21.0
1.0
31.
0.833…
6 5.000…
48
20
32.
1.166…
6 7.000…
6
10
6
40
e.
3
f.
4
9, , 0, 0.25, 3, 9.2, 100
5
d.
5
11, , 0, 0.75, 64
6
e.
5,
π
f.
5
11, , 0, 0.75, 5, , 64
6
π
f.
5, 0.3, 0, 2 , 4
37. The only whole number that is not a natural number
is 0.
38. Answers will vary. As an example, one integer that
is not a whole number is –3.
Section 1.3 The Real Numbers
43. Answers will vary. As an example, one number that
is an irrational number and a real number is
π
.
47.
5
32
since 3 is to the right of
5
2
1
22
.
48.
3
32
since 3 is to the right of
3
2
1
12
.
51. −2.5 < 1.5 since −2.5 is to the left of 1.5.
52. −1.25 < -0.5 since −1.25 is to the left of –0.5.
53.
35
44
since
3
4
is to the right of
5
4
.
54.
11
0 since 0 is to the right of
22
.
59.
0.3 0.3 since 0.3 0.333…
is to the right of 0.3.
60.
0.6 0.6
since 0.6 is to the left of
0.6 0.666…
61.
3.5 since 3.14
ππ
is to the right of −3.5.
63.
513
is true because
513
is true.
69.
17 6
is false because neither −17 > 6 nor −17 =
6 is true.
70.
14 8
is false because neither −14 > 8 nor −14 =
8 is true.
74.
99
because the distance between −9 and 0 on
the number line is 9 units.
75.
55
66
because the distance between
5
6
and 0 on
the number line is
5
6
units.
and 0 on the number line is
11
units.
78.
29 29
because the distance between
29
and 0 on the number line is
29
units.
Chapter 1 Variables, Real Numbers, and Mathematical Models
81.
3 0.6
5
0.6 0.6
0.6 0.6
0.6 0.6
Since
3
0.6 0.6, 0.6 .
5
83.
30 3 14 15
40 4 15 14
30 30 14
40 40
15
15
14
85.
13 13
813
1
13 8
1 1
1 1
Since
88
11, =1.
13 13
86.
44
2
88. rational numbers
89. integers
90. natural numbers