of 28
Chapter 7
RATIONAL EXPRESSIONS AND
EQUATIONS
10. 3
12. 4
32
x
x
−
−
2
3
x
=
14. 2
16
n
−
40
n
+=
or 4 0
n
−=
421
pp
−−
24210
24. and
yx yx
+−
5
y
+
()
()
(
)
()
34
34
52
10 2
rs
rs
()
()
224
Chapter 7 Rational Expressions and Equations
120
()
(
)
()
(
)
()
22 2
a b ab ab a b b
−−
(
)
()
(
)
2
32
75 75
75
31 31
3
yy yy
yy
yy y
yy
−−
−==
++
+
(
)
()
2
2
12
21
ab
ab
+
+==
()
()
()()
()()
22
22
1
yxyx
yxyx
+−
=− +−
=−
()
33
xx
−−−
()
()
(
)
()()
62
612 6
p
p
+
+==
(
)
(
)
()( )
34 4
xx x
+− +
()()
()()
()()
()()
25 36
65 2
6565
65
n
nn
nn
n
−
−+
=− −+
=− +
()
()
()
()()
()()
()
224
221
22
nnn
nn n
n
+−
+−
+
=+
()
2
3
3
2
xxy
xxy
xy
+
+=++
=+
()()
21518
2
23
ppqq
pq
pq
−+
+
=−
()
()
2
ab
+
()
33 4
(
)
(
)
()
2
25
310 5
xx
xx x
+−
−− −
()
22
331
hk hk
++
()
()
Section 7.2 Multiplication and Division of Rational Expressions
121
()()
225 0
550
m
mm
−=
+−=
88.
E
()
()
b. No
1
x
− are
equivalent whenever 1.x≠
clearly defined when 2
3
x
x
+
− is written as
Division of Rational
Expressions
23 2
33
8.
23
10 5
pq p q
⎜⎟
⎝⎠
10 2
pq ⋅
5
3
p q
t
()
43t+
ss
()
()
10 1
y
−
()
()
52y−
5
3
=
()
=61
10 21
x
xx
+
⋅
−+
()
()
26 1
10
x
x
+
=−
18.
()()
()
66 36
2210
ab ab
abab ab
⋅
−+
+− −
⋅
()()
()()
()
()()
()
()()
44xx+−
()()
()()
()()
()()
()
35 47
3
tt tt
t
=⋅
−+ −+
−
()
7t+
()
()
()
4t−
()
5t+
()
()
24.
22
⋅
53
aa
1
2
()
3
43
3
a−
=
()()
222
2
p
ppqppqq
pq pq
−−−
+−
q
()
32pq
+
s
−
v
÷=⋅
sss
s
3223
10 6 10 14
aabaab
−
10
a
()
()
()()
()()
42
tt
+−
Section 7.2 Multiplication and Division of Rational Expressions
123
()()
618 3 6 32 2
3
xx x x
++ + +
=
()
1
2x+
6
()
3x+
3x+
⋅
()
22x+
44.
84
÷
1
2
2
x
=−
()()
66
366
yy
yy y
+−
3y
=−
2
21 24
273 23
xxx
xx xx
++
++ +−
2
22
21 23
xxx
++−
=⋅
=
⋅
22 1
xx
+
()
2
+
()
()
2
22
2
xxy
yxxy
xy
+
=−⋅−
=− −
()
xx y+
⋅
52.
22
87 45
aa aa
÷
++ −−
()()
()( )
()()
()()
17 58
aa a a
=⋅
++ − −
()
()
()
()
54.
2
312
16
y
y
−
()()
()()
()
()()
()()
()()
()
()()
()
44 312
31 4 3 4
44312
yyyy
yy y
yy yy
=⋅
+− +−
++ −
=− ⋅
+− +−
=−
()
()
4y+
()
4y−
()
34y−
⋅
()
31y+
()
2
3
y
−
56.
22
98 67
xx xx
÷
++ −−
()()
()( )
()( )
()()
()
18 37
3
xx x x
x
=⋅
++ −−
−
=
()
8x+
()
1x+
()
8x+
()
1x+
⋅
()
7x−
()
3x−
()
7x−
1=
Chapter 7 Rational Expressions and Equations
124
()
4
h
+
()()
()
2
h
−
=−
()
4h+
4h+
()
1
2h
⋅
−
()
2
h
+
23
ba
2b
33
a2
2ab
62.
63
cd cd
⋅
−+
()
()
()()
222
32
cd cd cd
cd cd
++−
=⋅
−+
()
u
u
=⋅
=5u+
()
⋅
()
(
)
()
10 100
10
zx
−
=
(
)
100 dollars
zx−
()
21
gt gt
1
acac
bdbd
bd ad bc ac
⎝⎠⎝ ⎠
=− − +
72.
rMm
Mmm
m
=+
=⋅
+
=1M
Mm
m
⋅
+
a
⎛⎞
False
3. Answers may vary. Possible answer:
42
6
xx x
xx
+
+
2
39 2
xxx
−+
Section 7.3 Addition and Subtraction of Rational Expressions
125
of Rational Expressions
Exercises
2. 44
6
x
−
=
=
1
2
x
x
=−
12. 11 11
66 6
yy y
14. 8181
pq q p pq
++ +
sr
=− +
22.
()
21
xx
+
−=
24.
()()
()
51 5
aa
+−
−=
=23a+
2=
26. 58 43 58 43
+=
22
11
xx xx
++ ++
126
30.
()
222
2
54 54 54
24
aa aa aa
aaa
a
−+ −+ −+
++ −
−
=
()
()
1
a
−
()
4
2
a
a
−
−
32. 222
71 71 71
yy yy yy
−+
++ ++ ++
()()()
() ()
LCD 2 5 10
bbc bbc
=⋅⋅⋅ + = +
2
Factor 4 : 2
Factor 5: 5
ss
ss
⋅
++
()( )
()()( )
2
Factor 6 7 : 1 7
LCD 1 1 7
nn n n
nnn
+− − +
=+ − +
()()
()()()
()()()
2
Factor 2 9 10 : 2 5 2
LCD 2 5 3 2 or
25 3 2
xx x x
xxx
xxx
−+− −− −
=− + −
−− + −
22
3
Factor :
LCD 5
yy
y
=
55
y
yy
46. 2
Factor 4 : 2
xy x y
⋅⋅
()
2
23
43 12
xx
xy x xy
+⋅=
48.
() ()
Factor 3 : 3
cc c c
−⋅−
() ()
()
()
()
()
()()
LCD 2 1 1
yy
=+−
()()
()()
()
(
)
()
()()
1
222 1 1
y
yy
yyy
−
=⋅
++−
Section 7.3 Addition and Subtraction of Rational Expressions
127
()
()( )
()( )
2
LCD 1 4
tt
=− −
()
()
()
()( )
2
74
14
tt t
tt
tt
−
=
−−
()( )
()
()
()( )
41
tt
−
54. 32
4554
15 8
aa
+
23
95 37
57
75
45 21
c
c
cc
c
=⋅−⋅
22
22
68
qp
qp
pq pq
pq
=⋅+⋅
=
()()
22
2222
4
ab ab
abab abab
abab
abab
a
−+
=⋅+⋅
+− −+
−++
=+−
52
bb+−
()
92 5 7 5
52 2 5
bb
+−
=⋅− ⋅
=
64. 21
8
21 8
3
tt
−
−+
=−⋅
66.
2
4
2
y
y
−
()
2
2
22
227
2
4
2427
yyy
y
yyy
−−
=
−− +
128
68.
44
6
xx
x
−−
⎛⎞
70. ab
ab
+
−
=ab−
72. 43
()()
()
()
()
()()()
()
()
()
()
22
11
41 3
1
34 3
1
4
1
pp pp
pp pp
pp
pp pp
pp
pp
=−
−−
+−−+
=−
+−−−
=−
=− −
74.
532
11
532
11
nn
nn
nn
nn
−
+
−−
−
=− +
−−
2
1
32
4
p
pp
p
−+
78. 2
712
x
xx
++
()
()()
35
44 3
x
xx
−
=− ++
()()
()()
()()
()()
41
22 2
41
222
42 1
22 2 2
48 1
22
57
x
xxx
x
xxx
xx
xx x x
xx
xx
x
−
=−
−+−
−
=+
−+−
+−
=⋅+
−+ + −
++−
=+−
+
=+−
129
44 2
yy yy
−+ −−
()
()( )
() ()( )
()() ( )
()()
()()
()()
2
22
2
2
2
2
113 2
21
13 6
21
461
21
yy
y
yy yy
yy
yyy
yy
yy
yy
+−
−
−++ −
=
−+
−+ −
=
−+
−−
=
−+
31
4
y
−
()()
()()
()()
()()
31
4
224
y
yyy
−
=+
−−−
cc
c
+−
()()
()
()
()()
()()
()()
()()
()
()()
()()
()()
32 2 2
42 22
3
24 2 4 2 2
483 48
422
cc c c
c
ccc ccc
cc c c
cc c
−+−
=⋅ +⋅
+− +−
−+ −+
=+−
88. 2
21
xx xx
+− −−
()()
()()
()()()
()()
()()
()()
22
2
4
21 1
61 1 214
21 1
6712 4
21 1
8101
21 1
x
xx
xx xx x
xx
xx xxx
xx
xx
xx
−+−
−−+ +−
=+−
−++ +−
=+−
−+
=+−
()
y
−
()( )()()
()
()
2
2
34 3
283 4 28 3
224 1664824
82
25 26 40
yy y
yy y y y
y
yy
+−
−++−−−
=
−
−−
()()
()
2
Factor 2 : 2
LCD 2
yy
yy
−−
=−
Chapter 7 Rational Expressions and Equations
130
7
33
7
y
yy
y
⎝⎠
=+
−−
+
=−
()
()( )
24
64 2
35 2
648
a
aa
aa
aa
−
=+
−+ +
=++
−+ +
()()
()()
()
()
()()
()()
()()
()
()()
()()
()()
232 2
232 2
31
23
113 13 1
11
2
311
6332 1
311
63322
311
xx
xxx
xx xx xx
xx
xx x
xxx x
xx x
xxx x
xx x
−
=⋅+⋅
+− + −
+−
+⋅
+−
+−+ −
=+−
+−+ −
=+−
−
()()
()()()
LCD 224
jjj
=− + +
104.
ftf ftf
−+−
+=
106. The expressions are equivalent since
22 2
AAahAah
⋅−
108.
()
()
55 5
5
5
ssssss
ms m s
ss
−
−= ⋅−⋅
−− −
−−
=−
()
II I
+= and .
2
II
≠
Mindstretchers
()
()
31
1
41
1
41
xx
xx
x
xx
x
++
=+
+
=+
+
b. 2
51 3 2
11
1
x
xx
x
+=+
−+
−
2. 101n
131
the left is .qs Following the technique for
The right side is the sum of the rational
expressions on the left side.
Expressions
Exercises
2.
32
6
=÷
=
2
12
n
nn
−
()
21
1
n
n
−
=+
2
2
8
18 1
yy
yy
−
+−
()
81
yy
=−
8.
()
22
112
22
a
a
aa
a
=+
+
+
=
()
a
2a
⋅2a+
()
22aa=−
10. 222
22
4
4
ppq
qq
=−
−
2
pq q
+
()
()
q
12.
2102
10
t
tt
t
=+
+
()
()
23
25 1
t
t
tt
+
−
=+
14.
2
22
51 451
4
a
a
aaa
=−+
−+
()()
=2
a
a
⋅
()()
41 1aa−−
1
41
a
a
+
=−
Chapter 7 Rational Expressions and Equations
132
()
211
b
bb
−−−
221
1
221
1
21
1
23
1
b
b
b
b
b
b
b
b
−+
−
=−−
−
−
−
=−
−
18.
2
22
11 116
m
mm
mm
−−⋅
⎜⎟
⎝⎠
()
5
61
mn
m
=−
y
yyy
⎛⎞
() ()
()
()
()
()
()
2
2
2
2
2
2
515 2
2
536
yy
yy
yy
yy
yy
+
+
=++
+
+
=
++
322
dd
+−−
361
2
361
2
37
2
35
2
37 2
d
d
d
d
d
d
d
d
dd
−−
−
=−+
−
−
−
=−
−
−−
24.
x
=⎛⎞
()()
22
22
9
33
11
xx x
x
⋅−⋅
=
=3x−
()
33xx=+
22
2
2
2
1
1
32
1
32 1
bb
b
bb
b
bb
⋅− ⋅
−−
=−
+−
32
1
b
b
+
=+
Section 7.5 Solving Rational Equations
133
CD CD
()()
11
simplifies to .
CrDs CD
++ + +
30.
222
22
2
11
ppc
cc
mc
c
−⋅
−
⎜⎟
⎝⎠
⋅
32.
dD
sS sS
=
⋅+⋅
34. SSv
Sv f
f
−
1. Answers may vary.
3. Answers may vary. Possible answer:
4
Equations
2. 12
610 5
12
515
3
x
x
x
−=
⎛⎞
=
=
Check: 31 2
610 5
−=
4. 51 6
3
3
pp
p
=
2
23
yy y y
y
yy
⋅+⋅ =⋅
+=
134
20
2
y
y
−=
=
or 1 0
1
y
y
−=
=
8. 34
0
325
b
−
() ()
32 5
b
−
b
−
4
()( )
51512 8
bb
−= −
()
()
1340
32 5
440
55
440
−−−=
−−
−−=
−
() ()
1
43 3
3
x
x
x
+
=+
=
Check: 43
11
3
43
4
3
=
+
=
12.
33
43 2
3
ss
s
s
−−
−=
−
Check:
()
42
23 23
22
−=
−−
=
() () ()
()
2
2
2223
xx
xxx
−⋅ = −⋅ + −⋅
−
()
()
()()
2
560
230
xx
xx
−+=
−−=
20or 30
2
xx
x
−= −=
=3x=
Check:
() () ()
() ()
21
1
2
11
t
tt
tt tt
t
+
+⋅+ + 1
t
t
⋅+
()
() ()
2
22
1
21 1
22
2
tt
tttt
tttt
t
=+
++ = +
++ = +
=−
−
135
18.
()()
2
22
2
222
2
2
2
1
2
13
221
2
13
2221
2
62
02 6
023 2
xx
xx
xx
xxx
xx
xx
xx
xx
⎛⎞
⋅+=⋅
⎜⎟
⎝⎠
⋅+ ⋅= ⋅
+=
=−−
=+ −
() ()
131
44
41
4
11
+=
=
=
()()
253
25 3
22 22
y
y
y
yy yy
+=
+− +−
()() ()()
()()
()()
()
25
22 22
22
3
22 22
2
yy yy
yy
y
yy yy
y
+−⋅ ++−⋅
+−
=+ −⋅−+
+
()
2
22
yy
−⋅+
()()
22yy++ − 5
2y
⋅−
()
2y=+
()
2y−
()
3
2
y
y
⋅
−
()
2y+
()()
22523
245103
46
3
2
yyy
yy y
y
y
−+ +=
−+ + =
=−
=−
Check:
3
3
25 2
⎛⎞
−
⎜⎟
⎝⎠
−=
()
23 4
22
yyyy
+=
−−
() ()
()
()
()
23
22
2
4
22
yy yy
yy
yy yy
−⋅ + −⋅
−
=−⋅−
2y
()
y
=
()
2y−4
y
⋅
()
2y−
(
)
23 24
2364
510
2
yy
yy
y
y
+−=
+−=
=
=
Check:
() ()
2
23 4
22 2 22 2
23 4
02 44
+=
−−+
+=
−+
136
()
() ()
()
()
()
2
2
2
1
3
1
1
4
11
x
x
x
x
x
xx
+
=− + ⋅
+
+⋅
+
()
()
2
2
12
1
x
x
−+ ⋅
=− +
()
2
3
1
x
x
⋅
+
()
()()
22
2
442 213
442423
2320
21 20
xxxx x
xxxx x
xx
xx
+− ++=−
+− −−=−
+−=
−+=
210or 20
21 2
xx
xx
−= + =
==−
2
2
3
22
2
12 12
22 22
3
22
2
33
⎝⎠
−=−
⎛⎞
++
⎜⎟
⎝⎠
−=−
()
()
()()
()()
()()
()()
()()
()()
7
255
56
25 25
25 2
25
xxx
x
xx xx
x
xxx
xx
+−⋅
−
+
−+ −⋅+−
=+ −⋅
+
+− 7
⋅−
()()
()()
25
xx
⋅
+−
()
2x=+
()
52
x
xx
−⋅+
()( )()
2
2
7256 5
71456 5
xxxx
xxxx
+− += −
+−−= −
() ()
7561
61310 1
71 1
66
71 1
66
61
−+ −
−=
−+−
−− =−
−
−+=−
137
36
r
=
Check: 3112
36 4 36
111
+=
()
−
()
() ()
33
328
33
24
a
a
a
−=
−
−
−
()
8
2a−3
3
a−
⋅
()
1
2a−
24=
()
33
28
a−⋅
()()
1
8392
824918
6
aa
aa
a
−= −
−=−
−=
()
()
330
−
() ()
22
832 4
432
8
ssss
s
s
++=+
=−
=−
−
34. 1
24 16
48 48 48
24 16
2348
548
5
tt
tt
t
t
⋅+⋅=
+=
=
==
It will take the two crews 9.6 hr (or 9 hr 36
min) to do the job together.
36. 1
2
2
t
t
t
+=
=
138
38. 18 18 2
3
xx
+=
40.
()()
12
12 12 12
Ca
aA
aCa a
=+
+⋅=+⋅
+
42. a.
8
80 160
m
tt
m
tt
+=
+=
The speed was 2.4 MB per min.
1.
()
22
abab
ab
ab a b ab ab
⎜⎟
⎝⎠
+++=
22
,, and aab b are all positive numbers.
xy
131
xy
+=
4
431
5
43
y
+=
⎛⎞
3. Let d represent Diophantus’ age when he
died. Then, 6d is his age when his boyhood
is his age when his son was born, and
the LCD, 420.
139
2. 2
927
=
4. 22 44
35
x
=
35 70
6. 110
3
13 10
v
v
=
⋅=
10
()
75 75 600
75 675
75 675
p
p
p
−=
=
15 15
()
5303
nn
−=
Check: 15 6 15
()()
36918
4
tt
t
+=−
=
14.
2
16
416
n
n
=
⋅=
8
n
=−
8
n
=
Check:
48
−
=
48
=
140
()
()()
2
2
27
57 2
35 2
0235
075
x
xx
xx
xx
xx
+
⋅= +
=+
=+−
=+ −
18.
2
03 2
nn
=−−
32 1
2
3
nn
n
=− =
=−
Check:
2
13
2
22
33
3
−
=
−+
−
()
31 1 2
11
33
=
20.
( )() ()
()
22
2
2
33
23 33 3
23939
069
03
xx
xx xx
xx xx
xx
x
−−
+−= −
−−= −
=−+
=−
()
()()
2
239
aa
=+
230or 30
23 3
3
2
aa
aa
a
+= −=
=− =
=−
Check:
3
3
2
33
32
−
=⎛⎞
−+ −
⎜⎟
()
33
66
=
Section 7.6 Ratio and Proportion
141
504 9
v
=
99
()
45 18
45 45 720
45 675
45 675
t
t
t
+=
=
=
28. 48
=
96,000 of the 200,000 voters chose the
30.
()
275 165 2
275 165 2
xx
x
xx
xx x
+
=
=+
()
() ()
600 12,000 500 10,000
100 22,000
ss
s
−=+
=
36. 135
60 90
90 60 135
d
d
=
=⋅
38. 45
x
=
=
40.
60 120
2
x
x
+
=
=
=
2224x+=+=
142
2. If ,
bd
quantity bd to each side of the last equation,
6. The relationship is a joint variation.
10. 1
16.
()
35 7
ykx
k
=
=
3
8
k
=
20.
ykx
=
k
42
k
=
143
19
=
100
30.
54 1
6
yx
k
=
=
216
k
k
=
=
34.
ykxz
=
1.8
0.35
40.
()
60 2
60 2
yx
k
k
=
=
=
()()
250 0.04 100
k
k
=
=
144
50.
36 2
3
yx
k
=
=
()
64 2
64 4
64
4
k
k
k
=
=
=
1.9
4.75 ohms
145
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
2,
k
xz
=where 2
k is the constant of