Section 7.5 Solving Rational Equations
133
28.
()() ()()
11 11
1
CrDsCrDs
Cr Ds C D Cr Ds
++ + ++ +
=
++++
+++
CD
30.
2
222
22
11
m
mc
c
c
ppc
cc
⎛⎞
⋅
⎜⎟
⎝⎠
=⎛⎞
−⋅
−
⎜⎟
32.
()
dDsS
dD
dD dD
sS
sS sS
+⋅
+=⎛⎞
++⋅
⎜⎟
⎝⎠
Mindstretchers
1. Answers may vary.
2. Answers may vary.
7.5 Solving Rational
Equations
12
30 30 30
610 5
5312
515
3
x
x
x
x
⋅− ⋅ = ⋅
−=
=
=
42
10 5
22
55
=
=
6. 23yy
+=
23
yy y
y
⎛⎞
⋅+ =⋅
⎜⎟
⎝⎠
Chapter 7 Rational Expressions and Equations
134
6. (continued)
20
2
y
y
−=
=
or 1 0
1
y
y
−=
=
8. 34
0
325
b
b
−−=
−
34
b
−=
10.
() ()
43
1
4
113
1
43 3
31
1
x
xx
x
x
x
x
=
+
+⋅ = +⋅
+
=+
−=−
=
12.
43
2
33
43 2
s
ss
s
−=
−−
−=
Check:
()
32
42
23 23
46
2
11
−=
−−
−=
−−
3
00
Undefined
=+
3
11
963
99
=+
=+
=
16.
() () ()
21
1
2
11 11
t
tt
t
tt tt tt
+=
+
+⋅+ +⋅ = +⋅
Section 7.5 Solving Rational Equations
135
18.
2
22
2
13
1
2
13
221
2
xx
xx
xx
+=
⎛⎞
⋅+=⋅
⎜⎟
⎝⎠
2
Check:
2
13
1
33
222
+=
⎛⎞
⎛⎞
−−
⎜⎟
⎜⎟
20.
()()
2
253
22 4
25 3
22 22
y
yy
y
y
yy yy
−=
+− −
+=
+− +−
20. (continued)
()()
22523
245103
46
yyy
yy y
y
−+ +=
−+ + =
=−
25 2
34 43 916
22 22 4 4
9
25 2
−=
−+ + −
−
−=
()
()
()
4
22
2
yy yy
yy
=−⋅−
−2
2y
⋅−
()
3
2yy y
y
+−⋅
=
()
2y−4
⋅
Chapter 7 Rational Expressions and Equations
136
24.
()
2
43
2
11
xx
xx
−=−
++
() ()
22
4
112
1
x
xx
x
+⋅ −+⋅
+
Check:
2
2
11
43
22
2
11
11
22
3
22
2
12 12
22 22
⎛⎞ ⎛⎞
⎜⎟ ⎜⎟
⎝⎠ ⎝⎠
−=−
⎛⎞
++
⎜⎟
⎝⎠
−=−
⎛⎞
++
⎜⎟
26.
()()
2
756
52
310
756
525 2
xx
xx
xx
xx
xxx x
+
−=
−+
−−
+
−=
−+− +
10or 80
18
xx
xx
+= −=
=− =
Check:
(
)
() ()
2
51 6
71
15 12
13110
7561
61310 1
71 1
66
−+ −
−=
−− −+
−−−−
−+ −
−=
−+−
−− =−
−
Section 7.5 Solving Rational Equations
137
28. 3112
4
31 12
44
4
rr
rr
rr
+=
⎛⎞
⋅+ =⋅
⎜⎟
⎝⎠
30.
()
33
0
328
a
a
−−=
−
()
()
() ()
33
328
33
24 2 24 2
32 8
a
a
a
aa
a
−=
−
−
−⋅ = −⋅
−
32.
() () ()
81
4
8
44 41
4
s
ss
s
ss ss ss
ss
+=
+
+⋅+ +⋅ = +⋅
+
24 16
48 48 1
24 16
48 48 48
24 16
2348
548
48 9.6
tt
tt
tt
t
⎛⎞
⋅+=⋅
⎜⎟
⎝⎠
⋅+⋅=
+=
=
Chapter 7 Rational Expressions and Equations
138
38. 18 18 2
+=
40.
()()
12
12 12 12
aA
Ca
aA
aCa a
=+
+⋅=+⋅
+
42. a.
10 160
8
80 160
m
tt
m
tt
+=
+=
Mindstretchers
1.
11 1
11 1
abab
+=+
⎛⎞
If a and b are positive real numbers, then
22
,, and aab b are all positive numbers.
xy
(2) 43 3
xy
−+ =−
131
xy
+=
(1) 13
1
5
4
y
+=
431
5
y
+=
died. Then, 6d is his age when his boyhood
ended,
()
76dd+ is his age when he
married,
()
12 7 6ddd++ is his age when
his beard started growing,
()
12 7 6 5ddd+++
is his age when his son was born, and
()
12 7 6 5 2 4ddd d+++++ is his age when
Section 7.6 Ratio and Proportion
139
7.6 Ratio and Proportion
2. 2
927
227 9
x
x
=
⋅=
4. 22 44
35
22 35 44
x
x
=
=⋅
6. 110
3
13 10
310
v
v
v
=
⋅=
=
8.
()
75 30
20 1
75 1 20 30
p
p
=−
−= ⋅
8. Check: 75 30
20 9 1
75 30
20 8
15 15
=−
=
=
15
n
=
Check: 15 6 15
35
−=
624
66
4
t
t
−−
=
−−
=
Check: 39
=
()()
064
088
n
nn
=−
=+ −
80
8
n
n
+=
=−
or 80
8
n
n
−=
=
Chapter 7 Rational Expressions and Equations
140
16.
()
2
5
27
57 2
35 2
x
x
xx
xx
=
+
⋅= +
=+
18.
()()
2
2
1
32
23
03 2
032 1
n
nn
nn
nn
nn
=+
+=
=−−
=+ −
320or 10
32 1
2
3
nn
nn
n
+= −=
=− =
=−
Check:
20.
( )() ()
22
23 3
33
23 33 3
23939
xx
xx
xx xx
xx xx
+=
−−
+−= −
−−= −
22.
()
()()
2
2
2
3
32
233
239
2390
23 30
a
aa
aa
aa
aa
aa
=
+
=+
=+
−−=
+−=
230or 30
23 3
3
2
aa
aa
a
+= −=
=− =
=−
Section 7.6 Ratio and Proportion
141
24. 7
972
772 9
v
v
=
⋅=
26.
()
45 18
40 1
45 1 40 18
45 45 720
t
t
t
=+
+=⋅
+=
28. 48
100 200, 000
x
=
30.
275 165 2
275 165 2
xx
x
=+
+
=
32.
()
225 200
5
225 200 5
ss
ss
=
+
=+
100 22,000
220
s
s
=
=
The speed of the plane in still air is 220 mph.
3245
390
30
x
x
x
⋅=⋅
=
=
30 mix=
120 60 120
60 120
2
xx
x
x
=+
=
=
Chapter 7 Rational Expressions and Equations
142
Mindstretchers
1. Answers may vary.
2. If ,
ac
bd
= then .ad bc= If we add the
7.7 Variation
2. The number k is called the constant of
variation.
Decreases; inverse variation.
10. 1
Side perimeter
4
=
Increases; direct variation.
12. Tax Tax rate estimated value=×
Increases; direct variation.
18.
()
15 40
15
40
3
ykx
k
k
=
=
=
32
k
=
32yx=
22.
()
0.5 0.8
0.5
ykx
k
=
=
()
25 4
25 4
100
k
k
k
=
=
=
100
yx
=
Section 7.7 Variation
143
28.
k
yx
=
30.
k
yx
=
32.
()()
216 18 6
ykxz
k
=
=
34.
()()
120 16 15
120 240
120
ykxz
k
k
k
=
=
=
=
38.
()()
5.6 0.5 0.7
ykxz
k
=
=
()
3
28
60 16 16
kk
==
42.
()()
()
22
22
250
0.2 10
1000
yxz
k
k
=
=
=
22
1000
yxz
=
44.
2
kxz
yw
=
Chapter 7 Rational Expressions and Equations
144
46.
k
yx
=
48.
()()
90.63
ykxz
k
=
=
50.
k
yx
=
52. As the number of credit hours taken
increases, the total cost of the credit
hours increases.
Increases; direct variation
56. a.
R
kn=; the constant of variation
represents the price the company charges
for a tire.
R
56. b.
(
)
42 4000 168,000R==
$168,000
()
0.5 850
425
k
k
=
=
425
VP
=
60.
()
2
2
64 2
dkt
k
=
=
62.
()
6
2000
1.9 0.008
7.6 10
kL
RA
k
k−
=
=
=×
6
7.6 10
L
RA
−
×
=
Section 7.7 Variation
145
Mindstretchers
1. a. Answers may vary. Volume of a cylinder
varies directly with the square of the
radius and the height. The volume of a
2. a. Direct variation: y is doubled
Inverse variation: y is reduced by one-half
a
3. The statement “y varies inversely as x” can
be represented by 1,
k
yx
= where 1
k is the
constant of variation. The statement “x varies
inversely as z” can be represented by