Chapter 2
LINEAR EQUATIONS
AND INEQUALITIES
2.1 Solving Linear Equations:
The Addition Property
Exercises
2. Equations that can be written in the
4. To solve an equation means to find the value
that, when substituted for the variable, makes
the equation true.
6. Adding any real number to each side of an
equation results in an equivalent equation.
11 11
=
True
c.
()
()
()
35 18
35 1 18
34 18
12 18
x−=
−=
=
≠
False
10. Subtract 3 (or add 3).−
12. Subtract 15 (or add 15).−
14. Add 5.
16. Add 1
3.
4
18. 210
22 102
12
x
x
x
+=−
+−=− −
=−
20. 61
66 16
5
m
m
m
−=−
−+=−+
=
Check:
56 1
11
−=−
−=−
3
n
=
Check:
34 1
11
−=−
−=−
26. 25 21
25 21 21 21
46
y
y
y
−=+
−− =+−
−=
19 19
−=−
Chapter 2 Linear Equations and Inequalities
26
32. 14 6
14 14 6 14
8
c
c
c
−+=−
−++=−+
=
Check:
14 8 6
66
−+=−
−=−
36. 1
96
4
1
99 6 9
4
25 36
44
11
4
z
z
z
z
+=
−+= −
=−
=−
40. 3.4 9.5
3.4 3.4 9.5 3.4
6.1
r
r
r
−+=−
−++=−+
=−
Check:
()
3.4 6.1 9.5
9.5 9.5
−+− =−
−=−
42.
()
25 24
x
−− =
5
Check:
21 1
55 5
21 1
55 5
11
55
⎛⎞
⎟
⎜
−−− =−
⎟
⎜⎟
⎜
⎝⎠
−+=−
−=−
Section 2.1 Solving Linear Equations: The Addition Property
27
50. 11
21
72
n−=
15 3
72
n
−=
52.
(
)
()
5.2 12
5.2 5.2 12 5.2
17.2
x
x
x
+− =
+− + = +
=
Check:
17.2 5.2 12
12 12
−=
=
56. a
58. b
60. Add 3.
62. Check:
()
03232
032 32
32 32
−− =
+=
=
70. 2519 337
2519 2519 337 2519
2856 C
x
x
x
−=
−+=+
=°
Check:
2856 2519 337
337 337
−=
=
74. 70 90
70 70 90 70
20
x
x
x
+=
+−=−
=°
Check:
20 70 90
90 90
+=
=
28 Chapter 2 Linear Equations and Inequalities
Mindstretchers
1. a. The value of x decreases if a decreases and
b remains the same.
2.2 Solving Linear Equations:
The Multiplication
Property
Exercises
2. Multiply by 6.−
12. 88
88
88
1
y
y
y
−=
−=
−−
=−
Check:
()
81 8
88
−−=
=
16.
()
1.3
7
771.3
7
n
n
=−
⋅=⋅−
0.04
y
=−
Check:
(
)
2 0.04 0.08
0.08 0.08
−− =
=
20. 50 2
50 2
22
25
25
x
x
x
x
=−
−
=
−−
−=
=−
Check:
(
)
12
43
12
43
44
−−
=
=
=
33
Section 2.2 Solving Linear Equations: The Multiplication Property
29
28. 55
a
=
30. 28
7
72 7
8
27 2
z
z
−=
⎛⎞
⎟
⎜
−⋅− =−⋅
⎟
⎜⎟
⎜
⎝⎠
32.
() ()()
1.3
0.5
0.5 0.5 1.3
0.5
n
n
−=−
⎛⎞
⎟
⎜
−⋅− =−⋅−
⎟
⎜⎟
⎜
⎝⎠
36. 52
63
c
=
38.
() ()()
3.85
2.968
2.968 2.968 3.85
2.968
11.43
n
n
n
=−
−
⎛⎞
⎟
⎜
−=−⋅−
⎟
⎜⎟
⎜
⎝⎠
−
≈
Check:
(
)
89 72
72 72
−− =
=
Chapter 2 Linear Equations and Inequalities
30
48. 520
8
85 820
58 5
32
x
x
x
=
⋅=⋅
=
Check:
()
532 20
8
20 20
=
=
50. b
52. d
56.
() ()()
0.3
3.8
3.8 3.8 0.3
3.8
1.14
y
y
y
−=−
⎛⎞
−⋅− =−⋅−
⎜⎟
⎝⎠
=
Check:
1
10 10 2
10
20
t
t
⋅=⋅
=
Check:
()
120 2
10
22
=
=
20 hr
64. 1199.99
4
1
4 4 199.99
4
799.96
p
p
p
=
⋅=⋅
=
72
x
=
Check:
()
172 36
2
36 36
=
=
72 mi
Section 2.3 Solving Linear Equations by Combining Properties
31
70. 1
1 3000
2
r
=
72. 54 457
54 457
54 54
8.5
t
t
t
=
=
≈
74. 1
310
2
1121
3
332
x
x
=
⋅=⋅
Mindstretchers
1. The multiplication property of equality
allows you to divide each side of an equation
by any nonzero real number. So when you
divide 5x and 3x by x, you are assuming that
b. Answers may vary.
2.3 Solving Linear Equations
()
73 8 13
21 8 13
13 13
−=
−=
=
4
y
=
Check:
()
24 1 9
81 9
+=
+=
8.
24
3
22 42
3
s
s
s
−=−
−+=−+
Chapter 2 Linear Equations and Inequalities
32
10. 342
3
y
+=
12. 22
22 22
0
x
x
x
−=
−−=−
−=
14. 58
55 85
3
y
y
y
−=
−− =−
−=
16. 4313
5
433133
d
d
−=
−+= +
18.
314 10
8
y
+=−
24 14 10
10 10
−+=−
−=−
20. 43 21
aa
+=−
21 21
−=−
22. 418
318
xx
x
−=
=
=
24. 58 2 25
510 25
5510 255
xx
x
x
−− =−
−=−
−− =− −
Section 2.3 Solving Linear Equations by Combining Properties
33
26. 15
152
15 552
zz
z
z
−= −−
−= −
−− = − −
28. 36 6
36 6
nn
nn nn
+=−−
++=−+−
33
−=−
30. 753
73533
74 5
mm
mm mm
m
−=+
−− =+ −
−=
32.
()
324 2
3248
342448
tt
tt
tt tt
−= −
−= −
−−= −−
32. Check:
(
)
(
)
()
36 2 46 2
18 2 4 4
−= −
−=
22
9
a
−−
=−
Check:
(
)
(
)
()
59 4792
45477
−−= −+
−−=−
510
510
55
2
x
x
x
−=−
−−
=
−−
=
Check:
(
)
(
)
5232 4 32
−−=−
33
1
y
−−
=−
Check:
()
()
()
()
315 1 10 3 4 1 3 0
−+ − −+ =
Chapter 2 Linear Equations and Inequalities
34
40.
()( )
95 3 132
9515 132
14 15 15
nn n
nn n
nn
++=−+−
++=−−−
+=−−
42.
()()
52 2 4 25 3nn
⎡⎤
−−= −
⎣⎦
()
()
52 2 4 10 6
56 2 10 6
nn
nn
−+=−
−=−
44.
() ( )
742 3 11 2aa
⎡⎤
−+ − = +
⎣⎦
()
()
742 611 22
7221122
aa
aa
−+ −= +
−−+=+
44. Check:
()( )
() ()
74213 1112
7424 111
⎡
⎤
−+−− = −+
⎣⎦
⎡⎤
−+− =
48.
(
)
10.13 3.14 7.82
10.13 3.14 24.5548
10.13 3.14 3.14 3.14 24.5548
6.99 24.5548
6.99 24.5548
pp
pp
pp pp
p
p
=−
=−
−=−−
=−
−
16 15 31
31 31
+=
=
56. 311
r
+=
Section 2.3 Solving Linear Equations by Combining Properties
35
58.
()
763 6
76318
7363318
yy
yy
yy yy
−= −
−= −
−−= −−
60. 10 5 55
10 10 5 55 10
h
h
+=
−+ = −
62. 10 3 12 75
13 12 75
25 75
25 75
25 25
3
xx x
xx
x
x
x
++ =
+=
=
=
=
10 10 3 30x=⋅=
There are 30 lb of calcium in 75 lb of chalk.
66.
(
)
15 10 8 105
15 80 10 105
5 80 105
5 80 80 105 80
xx
xx
x
x
+−=
+− =
+=
+−= −
68. 1
30 40 4
30 40 10
tt
tt
⎛⎞
⎟
⎜
=−
⎟
⎜⎟
⎜
⎝⎠
=−
1.2 2400
1.2 2400
1.2 1.2
r
r
=
=
40 10 50 10
40 50 10 50 50 10
10 10 10
10 10 10 10 10
10 20
10 20
10 10
2
tt
tt tt
t
t
t
t
t
+= −
−+=−−
−+=−
−+−=−−
−=−
−−
=
−−
=
119
40 40 2 40 90
t
⎛⎞ ⎛ ⎞ ⎛⎞
⎟⎟⎟
⎜⎜ ⎜
+= += =
⎟⎟⎟
⎜⎜ ⎜
⎟⎟⎟
2. a.
The solution is 8.
The solution is between 4 and 6.
()
()
0 2 4 6 8 10 12
3260612182430
2 1 2 6 10 14 18 22 26
x
x
x
−−
+
36 Chapter 2 Linear Equations and Inequalities
2.4 Solving Literal Equations
and Formulas
Exercises
2. Literal equations can by solved for one variable
in terms of the others.
10. 5
5
5
ax b
ax b
aa
b
xa
=
=
=
5
2
x
y
−
=
18. 5210
522102
5102
5102
55
10 2
ab
abb b
ab
ab
b
+=
+−=−
=−
−
=
−
73
73
77
3
7
ac b
acb
bc
a
−=−
−−
=
−−
−
=
Section 2.4 Solving Literal Equations and Formulas
37
30.
2
2
R
A
s
Ra
=
32.
2
2
22
2
2
mv
Fr
rrmv
Fr
vv
Fr m
v
Fr
mv
=
⋅= ⋅
=
=
36.
()
1AP rt
APPrt
A
PPPPrt
=+
=+
−=−+
40. a.
2
ab
A
+
=
13
b
=
42.
A
lw
A
lw
ll
Aw
l
A
wl
=
=
=
=
T
CP P
P
CP T
⋅= ⋅
=
38 Chapter 2 Linear Equations and Inequalities
50. a. VBh=
52. a. 516
116
516
116 116 116
x
z
x
z
−
=
−
⋅= ⋅
54. a. 322Sl=−
b.
()
312 22
36 22
14
S
S
S
=−
=−
=
56. a.
Fma=
b. Fma
mm
=
Mindstretchers
2.5 Solving Equations
Involving Percent
6. 0.50 48 ft 24 ftx=⋅ =
8. 2.00 6 12x=⋅=
10. 0.06 9 0.54x=⋅=
0.1 $20
0.1 0.1
$200
x
x
=
=
20. 0.02 $5
x
⋅=
Section 2.5 Solving Equations Involving Percent
39
24. 0.0075 24
x
⋅=
65%
28. 10 12
12 10
12 10
12 12
0.833
x
x
x
x
=⋅
=
=
=
1
83 %
3
34. 0.1 8
80.1
80.1
88
0.0125
x
x
x
x
=⋅
=
=
=
1.25%
36. 0.19 $10,000 $1900x=⋅ =
40. 0.8 28
x
⋅=
21 7
21 21
0.333
x
x
=
=
1
33 %
3
46. 80.5
0.5 8
0.5 8
x
x
x
=⋅
=
54. 36
4
3
64
113
6
664
10.125
8
x
x
x
x
=⋅
=
⋅=⋅
==
Chapter 2 Linear Equations and Inequalities
40
60. 0.4 $240
0.4 $240
0.4 0.4
$600
x
x
x
⋅=
=
=
66. Cost in her town:
0.05 $250 $12.5⋅=
Cost across the river:
0.04 $250 $6 $10 $6 $16
⋅+=+=
She should buy the television in her town.
72.
(
)
(
)
$2000 $2000 0.04 3
$2000 $240
$2240
+
=+
=
0.2 $8000
0.2 0.2
$40, 000
x
x
=
=
$100,000 $40,000 $60,000−=
$40,000 gained 12%, and $60,000 lost 8%.
3
80.
(
)
(
)
(
)
(
)
10 0.05 0.02 10 0.03
0.5 0.02 0.3 0.03
0.5 0.5 0.02 0.3 0.5 0.03
0.02 0.2 0.03
xx
xx
xx
xx
+=+
+=+
−+ =−+
=− +
Mindstretchers
1. Answers may vary.
2. 1000abc
Section 2.6 Solving Linear Inequalities
41
3. b. The 5% increase is applied to the original
price. However, the 3% increase is applied
to a base price that includes the 5%
2.6 Solving Linear Inequalities
Exercises
2. A solution of an equality is any value of the
variable that makes the inequality true.
10. a.
()
21 10
23 1 10
61 10
710
x+<−
+<−
+<−
<−
False
10. d. 97611
11
97611
xx−≤ −
⎛⎞ ⎛⎞
⎟⎟
⎜⎜
−≤ −
⎟⎟
⎜⎜
⎟⎟
⎜⎜
18.
20.
22.
30. 51
55 15
4
x
x
x
−>−
−+>−+
>
Chapter 2 Linear Equations and Inequalities
42
36. 43
43 33
7
7
a
a
a
a
<−
+<−+
<
>
40. 721
721
77
3
t
t
t
−≤−
−−
≥
−−
≥
44. 2
3
332
3
6
x
x
x
−≥
⎛⎞
⎟
⎜
−⋅− ≤−⋅
⎟
⎜⎟
⎜
⎝⎠
≤−
50. 29 7
299 79
22
22
22
1
v
v
v
v
v
−>−
−+>−+
>
>
>
54. 24 9 13 8
24 24 9 13 8 24
91316
913 131316
416
ss
ss
ss
ss ss
s
−<− +
−−<− +−
−<− −
−+ <− + −
<−
67
67
77
77
77
1
yy
yy yy
y
y
y
−>−
−−>−−
−>−
−−
<
−−
<
58. 572 14
tt
−+ ≤−
Section 2.6 Solving Linear Inequalities
43
62.
(
)
0.2 10 5 9
d
−−>
64.
()()
24 3 3 4
86312
8663126
zz
zz
zz
−< +
−< +
−+< + +
68.
(
)
0.2 3 8
0.2 0.6 8
0.2 0.6 0.6 8 0.6
0.2 8.6
0.2 8.6
0.2 0.2
43
y
y
y
y
y
y
−>
−>
−+>+
>
>
>
72.
()()
34 2 23 6
12 12
2
mm
m
−≥ −
−−
≤
74. 211
4
333
zz z−<++
2.6 52
2.6 52
2.6 2.6
20
x
x
x
≤
≤
≤
78. b
80. d
Chapter 2 Linear Equations and Inequalities
44
88. 5644
mm
−+≥−+
90. 0.3 1200 2.8
0.3 0.3 1200 2.8 0.3
mm
mm mm
−<
−−< −
92. 12
14
14 14 12
14
168
A
A
A
≥
⋅≥⋅
≥
The area of the deck is at least 168 sq ft.
94. 1.5 1.38 3
ee
<+
96.
()
2412
2612
xx
x
+++≤
+≤
98. 52 20
72
t
t
+>
>
The flat fee is the better option if the customer
parks for more than 7 hrs.
The lowest score she can get is 9.5.
102. 10 36 12 10,000
4320 10,000
4320 4320 10,000 4320
5680
x
x
x
x
+⋅⋅≥
+≥
+−≥ −
≥
at least 5680 hours more
is true for 2,
a=− 1,b=− 3,c= and 5.d=
It is false for 2,a=− 1,b=− 5,c=− and