Chapter 7
RATIONAL EXPRESSIONS AND
EQUATIONS
7.1 Rational Expressions
Exercises
2. A rational expression is undefined if a
variable is replaced with a value that makes
its denominator 0.
8. 6
1
y
y
−
−
10
1
y
y
−=
=
14. 2
2
16
n
n
+
−
()()
216 0
440
n
nn
−=
+−=
16.
2
2
7
421
p
pp
+
−−
()()
24210
370
pp
pp
−−=
+−=
30
3
p
p
+=
=−
or 7 0
7
p
p
−=
=
18. 82 16
326yy
⋅=
24. and
yx yx
yx yx
−+
+−
The expressions are not equivalent.
32.
(
)
()
34
34
34 34
52
10 2
551
rs
rs
rs rs
==
34.
(
)
()
2
2
22 2
72
14 2
3
21 7 3
yz
yz
y
yz yz y
==
Chapter 7 Rational Expressions and Equations
120
38.
()
()
71
7
212
nn
nn
−=
−
46.
(
)
()
2
2
22
12
21
2
42 22
ab
ab
ba ab
+
+==
++
48.
(
)
()
25
10 2 2
55
mn
mn
nm mn
−
−==−
−−−
(
(
54.
()
33
1
33
xx
xx
−−
==−
−−−
56.
()
(
)
(
)
()
2
22
11
11
1
11
mm
mm
m
mm
+−
−−
==
+
++
64.
()()
()()
2
2
65 2
6710
56 56
25 36
65 2
nn
nn
nn
n
nn
−+
+−=−+
−
−+
()
()()
()()
()
32 2
23 23
221
31
22
3
nnn
nn n
nn n
nn n
n
n
+− +−
+−
=+−
+
=+
()
2
3
3
xxy
xxy
+
+=++
74.
()()
()
()
22
22
32
352
3
383 3
2
3
aba b
aabb
ab
aabb ab
ab
ab
−+
+− =+
+− −
+
=+
76.
(
)
33 4
74
4
5
55
ab ab
ab ab
−
−==−
Section 7.2 Multiplication and Division of Rational Expressions
121
84. 2
3
25
m
m
+
−
()()
225 0
550
m
mm
−=
+−=
88.
0
P
E
E=
The P/E ratio is undefined when the
company has no earnings.
92. a.
33
44
33
rr
ππ
π
==
Mindstretchers
1. The expressions
21 and 1
1
xx
x
−+
− are
equivalent whenever 1.x≠
7.2 Multiplication and
Division of Rational
Expressions
4. 52 5
23 2
xy x
⋅= 2
⋅5
33
yxy
=
6. 34
43 4
8
cd
dc
⋅=
1
c
3
d2
3d
⋅
84
2
c32
3
3
2cd
=
10.
()
43
412
t
tt t
+
+
⋅=⋅
14.
()
361013 2101
10 1
yy y y
y
⋅= ⋅
−− − −
−
=
()
32y−
()
52y−
⋅10 1y−
5
3
=
Chapter 7 Rational Expressions and Equations
122
18.
22
41010
ab ab
−−
⋅
20.
42
22
81 16
12 9
xx
xx x
−−
⋅
−− +
()()
()()
()()
22
2
99 44
34 9
xx xx
xx x
+− +−
=⋅
+− +
22.
22
22
421 20
215 328
tt tt
tt tt
+− +−
⋅
+− +−
()()
()()
()()
()()
()
37 45
35 47
3
tt tt
tt tt
t
−+ −+
=⋅
−+ −+
−
=
()
7t+
()
3t−
()
5t+
()
4t−
⋅
()
5t+
()
4t−
()
7t+
1=
3
28.
22
222
63
pq q pq
−+
⋅⋅
32.
2
222
ss sv
v
vvs
s
−−
÷=⋅
−
=2
v
v
⋅2
s
3
2
a
=
36. 2
5515
s
sss
s
÷= ⋅ =
38.
3223
2222
10 6 10 14
14
776
10
aabaab
ab
ab ab a b
−
÷=⋅
−
=−
5
3
a
14
⋅
2
ab
Section 7.2 Multiplication and Division of Rational Expressions
123
4
44.
2
484
84
xx+−
÷
()
()()
2
48 4
84
42 4
822
4
x
x
x
xx
+
=⋅
−
+
=⋅
+−
=
()
1
2x+
8
1
4
⋅
()
2
2x+
()
2
2
2
x
x
−
=−
48.
2
22
21 24
273 23
xxx
xx xx
++
÷
++ +−
2
22
21 23
xxx
++−
=⋅
() ()
()()
()
xx y
xy xyxy
xy
+
=− − ⋅ +−
=− −
()
xx y+
⋅
()
xy+
()
xy−
x=−
52.
22
22
56 13 40
87 45
aa a a
aa aa
−− − +
÷
++ −−
()()
()( )
()()
()()
22
22
56 4 5
87 1340
87 51
17 58
aa a a
aa a a
aa aa
aa a a
−− − −
=⋅
++ − +
−+ −+
=⋅
++ − −
()()
()()
()()
()()
()
()()
()
44 312
31 4 3 4
44312
31
yyyy
yy y
yy yy
y
+− +−
++ −
=− ⋅
+− +−
+
=−
()
4y+
()
4y+
()
4y−
()
34y−
⋅
()
31y+
()
2
y
−
Chapter 7 Rational Expressions and Equations
124
58.
()
2
2
284
4
hh h
h
+−
÷−
+
60. 23
99
23
ab
ba
⋅=
3
a
2
2b
b
⋅
33
1
a2
2
3
2ab
=
3
64.
2525
5
uu
uu
+
⋅
+
()
2
2
55
5
u
u
uu
u
+
=⋅
+
=5u+
()
55u+
⋅
u
5u=
70. a. ac
Cbd
=⋅
bd
=
72.
mM
rMm
=+
Mindstretchers
1. Answers may vary. Possible answer:
222
2
is 2 1, Q is 4 3, is 2 3,
and is 9
Px x x x Rx x
Sx
−+ −+ +−
−
2. Answers may vary. Possible answer:
a
aaacaac
b
b
cbc b bcb
⎛⎞
⎜⎟
⎝⎠
==≠
⎛⎞
Section 7.3 Addition and Subtraction of Rational Expressions
125
7.3 Addition and Subtraction
of Rational Expressions
Exercises
6. 4343
7
aa a
a
+
+=
=
8. 73 73
88 8
4
8
1
2
xx x
x
x
−−+
+=
−
=
=−
14. 8181
9
pq q p pq
pq
+
+=
++ +
=+
16. 10 14 10 14
4
rs sr sr
sr
−
−=
++ +
=− +
21
x
=−
24.
()()
()
51 5
51 5
2323 23
51 5
23
46
23
22 3
aa
aa
aa a
aa
a
a
a
a
+− −
+−
−=
++ +
+− +
=+
+
=+
+
=23a+
2=
Chapter 7 Rational Expressions and Equations
126
32. 222
2
2
43 6
71 71 71
43 6
71
23
71
yy y
yy yy yy
yyy
yy
y
yy
−−
−+
++ ++ ++
−− +−
=
++
+
=
++
38.
() ()
2
2
Factor :
Factor 4 : 2
Factor 5: 5
LCD 2 5 4 5
ss
ss
ss
ss ss
⋅
++
=⋅⋅+= +
40.
(
)
(
)
2
Factor 1: 1 1
nnn
−+−
LCD 2 3 12
xy xy
=⋅⋅⋅=
()
2
22
32
23
43 12
214
7
2
612
xx
xx
xy x xy
yy
y
xxy
+
+⋅=
⋅=
48.
() ()
22
Factor 3 : 3
cc c c
−⋅−
50.
(
)
(
)
()
()()
2
Factor 1: 1 1
Factor 2 2 : 2 1
LCD 2 1 1
yyy
yy
yy
−+−
++
=+−
()()
2
442
112
1
8
yy
yy
y
y
=⋅
+−
−
=+−
Section 7.3 Addition and Subtraction of Rational Expressions
127
52.
()
()( )
()( )
2
2
2
2
Factor 2 1: 1
Factor 5 4 : 1 4
LCD 1 4
tt t
tt t t
tt
−+ −
−+ − −
=− −
54. 32
45
35 24
4554
15 8
20 20
23
20
aa
aa
aa
a
+
=⋅+⋅
=+
=
35
c
58.
22
22
68
68
pq pq
qp
+
=⋅+⋅
60.
22
22
ab ab
ab ab
abab abab
+
+−
−+
=⋅+⋅
+− −+
()
92 5 7 5
52 2 5
2182535
10 10
2182535
10
2182535
10
37 7
bb
bb
bb
bb
b
+−
=⋅− ⋅
+−
=−
+− −
=
+−+
=
−
=
8
=
66.
2
2
22 7
4
2
yy
y
y
−−
−
Chapter 7 Rational Expressions and Equations
128
68.
()
6
44
6
44
x
xx
x
xx
−
−−
⎛⎞
=−
⎜⎟
⎜⎟
−−−
1=
72. 43
1
pp
pp
++
−−
41 3
11
pp pp
pp p p
+− +
=⋅−⋅
−−
74.
532
11
532
11
nn
nn
nn
nn
−
+
−−
−
=− +
−−
76.
()( )
2
4
1
32
4
12 1
p
p
pp
p
pp p
+−
−+
=+
−− −
()() ()
()()
245 3
43444 3
8515
44 3
315
xx
xx x x
xx
xx
x
−+
=⋅+⋅
++ + +
−+ +
=++
−+
=++
()()
()()
22 2 2
48 1
22
57
xx x x
xx
xx
x
=⋅+
−+ + −
++−
=+−
+
Section 7.3 Addition and Subtraction of Rational Expressions
129
82. 22
13
44 2
yy
yy yy
−+
−+ −−
()
()( )
2
13
12
2
yy
yy
y
−
=+
+−
−
84. 2
31
4
268
y
yyy
−
+
−−+
86. 2
32
24 4
cc
cc
c
++
+−
()()
()
()
()()
()()
32
= 24 2 2
42 22
3
24 2 4 2 2
cc
cccc
cc c c
c
ccc ccc
++
++−
−+−
=⋅ +⋅
+− +−
88. 2
61 4
21 1
21
xx x
xx xx
−+−
+− −−
()()
61 4
21 121 1
xx x
xx xx
−
=+−
+− +−
34 3
yy y
+−
()
()
()
()
2
22
2
22
2
2
82
283 28824
82
224 1664824
82
y
yy yy y
y
yyy y y
y
=
−
−+ −−−+
=
−
−+ − −−+
=
−
Chapter 7 Rational Expressions and Equations
130
96. 2
24
315
710
a
a
aa
−++
++
()()()
()
24
5235
2
234
52335 2
a
aa a
a
a
aa a a
−
=+
++ +
+
−
=⋅+⋅
++ + +
98. 2
22
13
1
xx
xx
x++
+
−
()()
()()
()
()
()()
22
11 13
31
23
113 13 1
311
xx
xx x x
xx
xxx
xx xx xx
xx x
=++
+− +
−
=⋅+⋅
+− + −
=+−
100. 52 2 522
3131 31
32
31
xxxx
xx x
x
x
−−−
−=
−− −
−
=−
1
t
==
106. The expressions are equivalent since
22 2
.
AAahAah
a
hhhh
⋅−
−= − =
The bus took
()
5
ss− hr longer than the
train.
110. The expressions are not equivalent since
HW HW
II I
+
+= and .
2
HW HW
II
++
≠
b. 2
51 3 2
11
1
x
xx
x
+=+
−+
−
2. 101n
n
+
Section 7.4 Complex Rational Expressions
131
3. a. Answers may vary.
b. The LCD of the rational expressions on
the left is .qs Following the technique for
7.4 Complex Rational
Expressions
Exercises
2.
3
2
32
3
3
2
s
s
ss
=÷
4.
1
11
12
n
nn
n
nnn
−
−+
=÷
+
6.
2
22
11
1
181
8
y
yy
y
+
+
=−
−
8.
2
44
112
a
aaa
a
−
−
=+
10. 222
22
2
2
4
4
ppq
qq
ppq
qq
+
+
=−
−
22
2
2
22
24
2
4
pqp q
qq
pq q
qpq
+−
=÷
+
=⋅
−
12.
22
23 23
2102
t
ttt
t
−
−
=+
()
25 1
tt
14.
2
22
2
11
1
51 451
a
aa
a
a
−
−
=−+
Chapter 7 Rational Expressions and Equations
132
16.
(
)
()
211
1
211
1211
211
b
bb
b
bb
−+
+−−
=−−
−−−
18.
2
2
22
6
23
23
11 116
nn
nn m
m
mm
mm
⎛⎞
+⋅
+⎜⎟
⎝⎠
=⎛⎞
−−⋅
⎜⎟
⎝⎠
20.
()
()
52
5
5
3352
22
y
yyy
yyyy
yy
yy
⎛⎞
⋅+
⎜⎟
⎝⎠
=⎛⎞
++⋅ +
⎜⎟
++
⎝⎠
22.
()
()
321
1
322
1321
322
d
dd
d
dd
−−
−−−
=−+
+−−
24.
2
22
11 11
3
33
x
x
xxxx
⎝⎠
=⎛⎞
−−⋅
⎜⎟
⎝⎠
22
9
33
11
xx x
x
⋅−⋅
=
()
33xx=+
26.
2
2
2
2
22
12
12 3
3
11
11
b
bb
bb
b
⎛⎞
−− ⋅
−− ⎜⎟
⎝⎠
=⎛⎞
−−⋅
⎜⎟