Section 1.4 Basic Rules of Algebra
95. a.
97. – 107. Answers will vary
108. makes sense
109. does not make sense; Explanations will vary.
110. does not make sense; Explanations will vary.
111. makes sense
112. false; Changes to make the statement true will vary.
A sample change is:
2
3
is an example of a rational
number that is not an integer.
116. true
117. false; Changes to make the statement true will vary.
A sample change is: All integers are rational
numbers.
120.
31.732
and should be graphed between 1
and 2.
124.
3( 5) 3(4 5) 3(9) 27
and
3153(4)15121527
x
x
Both expressions have the same value.
and
126.
9 2 9(4) 2(4) 36 8 28
and
77(4)28
xx
x
Both expressions have the same value.
2. a.
14 14
xx
b.
77yy
3. a.
517175
xx
Chapter 1 Variables, Real Numbers, and Mathematical Models
5.
8( 4) 8(4 )
(8 4)
12 or 12
xx
x
xx
9. a.
87103(810)(73)
18 10
xxxx
x
b.
9 6 5 2 (9 5 ) (6 2 )
14 8
xyxy xx yy
xy
1.4 Concept and Vocabulary Check
1. like
2.
ba
3. ab
1.4 Exercise Set
1. 3x + 5
3. x + 2 + 5x
a. 3 terms
b. 1
c. 2
d. x and 5x are like terms.
d. no like terms
6. 8y + 1 + 10x
a. 3 terms
b. 8
c. 1
d. no like terms
7. 44yy
Section 1.4 Basic Rules of Algebra
14. 6( 4) 6(4 )xx
15.
99 or 9
xx x
21. 5( 3) ( 3)5xx
22. 6( 4) ( 4)6xx
23. 7(5 ) (75) 12xxx
29. 8(23)8(2)8(3)1624xx x
33.
555xy x y
34.
777xy x y
35.
323 3236xx x
39.
111
512 5 12
222
56
xx
x
43.
6326 6362
61812
xyx y
xy
44.
72 4 72 74 7
14 28 7
xy x y
xy
48.
513 513 18xx x x
53.
536 56 3113yyyy y
54.
8710 810 7187yyyy y
55.
257427 54
91
xx xx
x
Chapter 1 Variables, Real Numbers, and Mathematical Models
58.
13 15 2 11 13 2 15 11
15 26
aa aa
a
63.
73 2 54 2
ab ab
21 14 20 10
21 20 14 10
41 24
abab
aabb
ab
65.
72( 9)
7 (2 18) Distributive Property
x
x
66.
5( 4) 3
(5 20) 3 Distributive Property
xx
xx
67.
72xx
72 9xx x
70.
11 5xx
11 5 6xx x
9(3 ) 27 9xx
75. 85( 1)x
85(1)85553xxx
76. 93( 2)x
10 137
n
This is the exact value shown in the bar graph.
78. a.
23
3(50 95) 2 4
Wn n
1850
According to the formula, there were 1850
million worldwide internet users in 2009.
Mid-Chapter Check Point
89. does not make sense; Explanations will vary.
Sample explanation: Subtraction does not have a
commutative property.
91. makes sense
92. makes sense
93. false; Changes to make the statement true will vary.
A sample change is: (24 6) 2 24 (6 2).
97. 60 because
150 90 60
98.
60
because
50 10 60
99.
5
because
30 35 5
Mid-Chapter Check Point – Chapter 1
1.
210 210(6)
260
x
3.
310
4( ) 4(10 3)
22
30 4(7)
xy yx
4.
12
4x
3(6 2) 4 6 1
3(8) 24 1
24 23, false
The number is not a solution.
1
The number is not a solution.
8. a.
232
2(8) 32
48
Cn
The average price of expanded basic cable
service in 2008 was $48.
This underestimates the actual number shown in
the bar graph by $2.
b.
232
2(20) 32
72
Cn
10.
23 2
34 3
3
21
442
Chapter 1 Variables, Real Numbers, and Mathematical Models
12.
39 3103253
510 59 533
5
25
3
2
3
3
13.
23 2 21 3
105 105 105
7
357
1
5
16.
5(32)
9
55
(50 32) (18) 10
99
CF
C
50 F
is equivalent to
10 C.
21. 5( 3) ( 3)5xx
22. 5( 3) 5(3 )xx
23. 5( 3) 5 15xx
1.5 Check Points
1.
4(7) 3
Start at 4 and move 7 units to the left.
3. a. 10 ( 25) 35
b.
0.3 ( 1.2) 1.5
b.
3 ( 10 ) ( 10 ) 16
3 ( 10 ) ( 10 ) 16
3 ( 10) ( 10) 16
76
yz yz
yy zz
yz
yz
Section 1.5 Addition of Real Numbers
1.5 Concept and Vocabulary Check
1. additive inverses
2. zero
1.5 Exercise Set
1.
734
2.
725
4.
15 6
8.
550
15.
810 18
16.
46 10
17.
0.4 0.9 1.3
18.
1.5 5.3 6.8
21.
94 5
22.
73 4
23.
12 8 4
Chapter 1 Variables, Real Numbers, and Mathematical Models
31.
939 63
10 5 10 10 10
34.
51 52 3 1
636662
35.
3 4 15 28 43
7 5 35 35 35
38.
10 3 8 10 3 8
781
41.
17 4 2 3 10
13 2 3 10
15 3 10
18 10
8
42.
19 5 1 8 13
14 1 8 13
7
20 7
20 1
21
45.
3.5 45 8.4 72
3.5 8.4 45 72
50.
26 14 26 14 12yy yy
51.
815 815
23
aa a
a
52.
913 913
22
aa a
a
Section 1.5 Addition of Real Numbers
53. 4(13)(10)17
4 ( 10 ) ( 13 ) 17
64
yz yz
yy zz
yz
56. 10 13 ( ) ( 4)
10 ( ) 13 ( 4)
11 9
bb
bb
b
57. 7(5)(9)19
7(9)(5)19
214
xy xy
xx yy
xy
60. 7(3 5) ( 25 )
21 35 ( 25 )
21 ( 25 ) 35
435
yy
yy
yy
y
63.
20 15 25
20 10
20 10
30
64.
25 18 26
25 8
25 8
66.
8610 892
14 10 8 7
4 1
41
67. 6(13)xx
6(13) 19xx x
70. 15 4
xx
15 4 11
xxx
71. 56 100 44
level.
75. 7155 3
The temperature at 4:00 P.M. was 3°F.
Chapter 1 Variables, Real Numbers, and Mathematical Models
76. 15 13 ( 4) 6
The team had a total loss of 6 yards.
78.
20 3 ( 2) ( 1) ( 4) 2
20 3 2 2 ( 1) ( 4)
25 7
18
The level of the reservoir is 18 feet.
79. a. 2303 ( 3603) 1300
The deficit in 2011 was $1300 billion.
80. a. 2568 ( 2729) $161
The deficit in 2007 was $161 billion.
b.
2105 ( 3518) $1413
81. – 87. Answers will vary.
88. makes sense
92. false; Changes to make the statement true will vary.
A sample change is:
sum equals the difference of their absolute values.
95. false; Changes to make the statement true will vary.
A sample change is: The sum of a positive number
and a negative number is sometimes negative.
96. The sum is negative. The sum of two negative
numbers is always a negative number.
97. The sum is negative. When finding the sum of
of the sum. Since c is further from 0 than b, we use
a positive sign.
99. Though the sum inside the absolute value is
negative, the absolute value of this sum is positive.
b. 0, 4
c. 6, 0, 4
Section 1.6 Subtraction of Real Numbers
104.
16 2( 1)
16 2(18 1) 18
xx
1.6 Check Points
1. a.
311 3(11) 8
b.
4(5) 459
3.
10 ( 12) 4 ( 3) 6
10 12 ( 4) 3 ( 6)
(10 12 3) ( 4) ( 6)
25 ( 10)
15
4.
64 7aab
has terms of
6,4,a
and
7.ab
5. a.
42 9 4(29)
xx x
b.
115 11514
5. 3; (12); (23)
6. (4y); 6
7. three; addition
3. a. 7
b.
5757
4. a. 8
b.
2828
5.
14 8 14 8 6
6.
15 2 15 2 13
Chapter 1 Variables, Real Numbers, and Mathematical Models
13.
13 2 13 2 11
17.
45 45 45 45 0
18.
65 65 65 65 0
19.
23 23 23 23 0
20.
26 26 26 26 0
26.
0 15 0 15 15
27.
35 3 5 2
77 7 7 7
28.
47 4 7 3 1
99 9 9 9 3
29.
13134
55555
55 555
34.
41 4131
9 9 9993
37.
111 1 2 1 1
2424444
38.
21 2 1 4 1 3
5 10 5 10 10 10 10
39.
9.8 2.2 9.8 2.2 7.6
44.
1.4 1.4 1.4 1.4 2.8
45.
2.06 2.06 2.06 2.06 0
46.
3.47 3.47 3.47 3.47 0
47.
52 5 2 3
πππ π π
48.
97 9 7 2
ππ π π π
49.
31031013
πππππ
Section 1.6 Subtraction of Real Numbers
52.
14 3 7 14 3 7
14 7 3
21 3
18
55.
62310
62310
62103
18 3
15
58.
63811
638 11
61138
17 11
6
59.
23 11 7 25
23 11 7 25
23 11 25 7
59 7
823 146 50 832
823 146 50 832
1019 832
187
62.
726 422 921 816
726 422 921 816
632
11 4
66
71
or 1
66
Chapter 1 Variables, Real Numbers, and Mathematical Models
65.
0.16 5.2 0.87
0.16 5.2 0.87
0.16 5.2 0.87
5.36 0.87
4.49
67.
31 5 3 1 5
44 8 4 4 8
45
48
85 3
88 8
210 8
xx
75.
47 17 47 17
47 17
410
yy y y
y
y
29 5
75 or 57
a
aa
78.
3711 37 11
3117
311 7
aaa a
aa
a
Section 1.6 Subtraction of Real Numbers
82.
15 3 8 10xx
15 3 8 10
15 10 3 8
25 11
xx
xx
x
84. 69415xyx y
69 415
64 915
64 915
10 6 or 6 10
xy xy
xx yy
xy
xy y x
51
84
51
84
512
10 4 10 10 4 5 10 2
9514
10 20 20
95 14
10 20 20
99
90.
24 16 51 31 2
24 16 51 29
24 16 51 29
40 22
40 22
18
Chapter 1 Variables, Real Numbers, and Mathematical Models
91. 6(5)xx
6(5)6511xxxxx
94.
12 7
xx
12 7 12 7 5
xx xxx
95. Elevation of Mount Kilimanjaro – elevation of
Qattara Depression
19,321 436 19,757
The difference in elevation between the two
geographic locations is 19,757 feet.
97. 10 ( 15) 5
shrink by 5 years
98. 2(5) 3
shrink by 3 years
99. 5(6) 11
shrink by 11 years
105. 5(6) 561
1 year
109.
2000 400 500 2100
–2100 meters
110.
900 300 200 800
–800 meters
111. – 115. Answers will vary.
116. makes sense
117. makes sense
A sample change is: 7(2) 72 9
122. true
123. true
124.
(positive number) (negative number)
(positive number) (positive number)
ca
ca
127.
0 0 (negative number)
0 (positive number)
positive number
b
b
129. The calculator verifies your results.
130. The calculator verifies your results.
131.
13 3 3(5 1)
13 2 3 3(5 2 1)
26 3 3(10 1)
29 3(9)
29 27, false
xx
The number is not a solution.
136. 2( 3) 6
1( 3) 3
0( 3) 0
1.7 Check Points
1. a.
8( 5) 40
b.
14 4
b. The multiplicative inverse of 1
8 is 8 because
181.
8
c. The multiplicative inverse of 6 is 1
6
because
1
61.
6
22
5. a. 32 8
4
Chapter 1 Variables, Real Numbers, and Mathematical Models
d.
00
5
6. a. 4(5 ) ( 4 5) 20xxx
7.
4(3 7) (13 2) 12 28 13 2
12 13 28 2
126
26
yyyy
yy
y
y
1.7 Concept and Vocabulary Check
1. positive
2. negative
3. negative
4. positive
10. undefined
11. positive
4.
95 9545
5.
37 21
13.
3312
12 9
441
14.
4430
30 24
551
15. 343412
575735
Section 1.7 Multiplication and Division of Real Numbers
18. 52 52 10
11 7 11 7 77
19.
31.2 3.6
20.
41.2 4.8
26.
2713 42
27.
23 4 1 24
28.
32 5 1 30
33.
8401760
34.
91218030
35. The multiplicative inverse of 4 is 1
4.
40. The multiplicative inverse of −12 is 1
12
.
41. The multiplicative inverse of 2
5
is 5
2
.
6
45. a. 60 1
60
55
b. 1
60 12
5
48. 40 1
40 8
55
49. 21 1
21 7
33