Section 7.5 Complex Rational Expressions
34.
1
2
1
12
x
x
−
−−
LCD =
2x−
()
() ()
1
211
2
121 3
21 2
xx
xx
xx
−
−
==
−− −
−−
−
36.
7
9
4
6
xx
xx
+−
−+
LCD = x
2
2
7
997
464
6
xx xx
x
xx
xx x
+−
+−
=
−+
−+
38.
34
3
25
53
xy xy
xy
xy
+
−
LCD =
34
xy
34
34 32
323
34
25
25
53
53
xy
xy xy yx
yxy
+
+
=
−
()()
()()
2
2
2
1
3333
11
5
1
6
11
5
x
xx
xx
x
xx
−
−− −
+−
=
−
−
+−
=
Chapter 7 Rational Expressions
40.
()()
2
33 33
22 22
55
22
4
xx xx
xx
x
−−
+− +−
=
+−
−
41.
()()
2
61
61
53 3
3
215
11
11
55
xx x
x
xx
xx
−
−+− −
−
+− =
++
++
LCD =
()()
53xx+−
42.
()()
()()
2
16
16
225
2310
11
11
22
LCD 2 5
xxx
xxx
xx
xx
−
−−−+
−+− =
++
−−
=− +
Section 7.5 Complex Rational Expressions
43.
()
()
1
1
11
55
55
LCD 5
yy yy
yy
−
−
−
−+ +
=
=+
44.
()
()
1
1
11
22
22
LCD 2
yy yy
yy
−
−
−
−+ +
=
=+
46.
()()
()
11
1111
1111
1111
111
11
11
xxxxx
xxxx
x
xx
xx
xx
++++
−= −= −= −= −
+−
−+−
++
+−
==
Chapter 7 Rational Expressions
49.
12
2d
dd
rr
+
LCD =
12
rr
50.
12
2121
21
12 12 12 12
111
11
R
R
RRRR
R
R
RR RR RR RR
===
++
++
Let
12
10 and 20,RR==
12
21
10 20 200 20 ohms
20 10 30 3
RR
RR
⋅
===
++
b. According to the data in the bar graph, 4.6
6.9 or
about 67% was revenue from digital sales in 2013.
c.
22
22
10 135 (6) 10(6) 135
9 85 525 9(6) 85(6) 525
xx
xx
++ + +
=
−+ − +
52. a.
2
2
2
2
22
10 135
10 135 1
1
85 525 85 525
99
xx
xx
x
x
xx
xx
++
++
=
−+ −+
174 0.50
351
=≈
According to the model about 50% was revenue
from digital sales in 2010.
This overestimates the actual percent by 1%.
53. – 55. Answers will vary.
.
xy
−
60. false; Changes to make the statement true will vary.
A sample change is: It is not complex because it
does not contain any secondary fractions.
Section 7.5 Complex Rational Expressions
63. false; Changes to make the statement true will vary.
A sample change is: All complex rational
expressions can be simplified by both methods.
64. Simplify the given complex fraction.
LCD =
6
x
Because the rational expression can be simplified to
3
,x this is what it does to each number x; it cubes x.
66.
2
22
16 1
11
56 23
11
yyy
yy
yy
+− −
−
−+ −−
Simplify the first complex rational expression using
the LCD method.
22
22
2
1
1
23
23
1
y
yyy
yyy
yy
−
−
⋅=
−−
−−
()()
()()
22
43
31
yy yy
yy
++− −
=−+
1
111
11
1
xxxx
xxx
xx x
x
++
==
+
=⋅=+
Change the expression on the right to
1.x+
Chapter 7 Rational Expressions
70.
32
22050xxx
−+
()
()
2
2
21025
25
xx x
xx
=−+
=−
72.
()
()
22
xyx xyy+−+
()()
2222
32 22 23
33
xx xy y yx xy y
xxyxyxyxy y
xy
=−++−+
=− + + − +
=+
74.
214
332
214
66
332
xx
xx
=−
=−
75.
2
2
225
2520
(2 1)( 2) 0
xx
xx
xx
+=
−+=
−−=
210 or 20
xx
−= − =
1
668
1
24 24
668
1
24 24 24
668
443
xx
xx
xx
xx
=+
=+
⋅= ⋅+ ⋅
=+
Section 7.6 Solving Rational Equations
2.
5171
2183xx
=−
The restriction is
0x≠
.
The LCD is
18 .x
3.
65xx
+=−
The restriction is
0x≠
.
The LCD is .x
4.
2
11 4 3
55
25
xx
x
+=
+−
−
Chapter 7 Rational Expressions
5.
39
33
x
xx
=+
−−
The restriction is
3.x≠
The LCD is
3.x−
39
33
x
xx
=+
−−
6.
250
100
x
yx
=−
The restriction is
100.x≠
250
750 100
x
x
=−
Section 7.6 Solving Rational Equations
7.
()
2
(2) (2) 2
b
ax
b
xax x
=+
+=+
8.
11 1
xy z
+=
11 1
{
}
xyz xyz
xy z
+=
7.6 Concept and Vocabulary Check
1. LCD
2. 0
3. 8x
10. false
7.6 Exercise Set
1.
2
32
xx
=−
6662
32
2312
012
12
xx
xx
x
x
⋅=⋅−⋅
=−
=−
=
30 30 30 1
56
6530
30
xx
xx
x
⋅= ⋅+ ⋅
=+
=
The solution set is
{
}
30 .
3.
4
xxx
=−
Chapter 7 Rational Expressions
4
18 18 18
{
}
xxx
⋅=⋅−⋅
4.
5
4122
xxx
=−
There are no restrictions.
The LCD is 12.
5
{
}
{
}
xxx
=−
5.
8
26
x
−=
The restriction is
0x≠
.
The LCD is x.
8
26
{
}
{
}
x
−=
6.
9
14
−=
9
14
94
93
3
xx x
x
xx
x
x
⋅− ⋅ = ⋅
−=
−=
−=
333
3
612
6
xxx
xx
x
x
⋅+ ⋅= ⋅
+=
=
The solution set is
{
}
6.
Section 7.6 Solving Rational Equations
9.
2513
324xx
+= +
The restriction is
0x≠
The LCD is 4x.
{
}
10.
7522
23 3xx
=+
The restriction is
0x≠
11.
21111
3463xx
+= −
The restriction is
0x≠
.
The LCD is 12x.
21111
{
}
+= −
12.
581 1
29183xx
−= −
The restriction is
0x≠
.
The LCD is 18x.
The solution set is
{
}
3.
13.
64
33xx
=
+−
15
x
=
The solution set is
{
}
15 .
14.
75
13xx
=
+−
Restrictions:
1, 3xx≠− ≠
Chapter 7 Rational Expressions
15.
21
1
2
xx
xx
−+
+=
Restriction:
0x≠
LCD = 2x.
16.
7494
55
x
xx
−=−
Restriction:
0x≠
LCD = 5x.
17.
67xx
+=−
Restriction:
0x≠
LCD = x
{
}
67
xx
+=−
18.
78xx
+=−
Restriction:
0x≠
LCD = x
7 1
xx
=− =−
The solution set is
{
}
7, 1 .−−
19.
50
5
x
x
−=
5 0 or 5 0
5 5
xx
xx
+= −=
=− =
The solution set is
{
}
5, 5 .−
20.
40
4
x
x
−=
Section 7.6 Solving Rational Equations
21.
312
xxx
+=
Restriction:
0x≠
LCD = x
312
xxx
+=
22.
319
xxx
+=
Restriction:
0x≠
LCD = x
319
xxx
+=
23.
47
22
y
y−=
Restrictions: 0y≠
LCD = 2y
47
22
y
y
−=
24.
41
33
y
y−=
Restrictions: 0y≠
LCD = 3y
41
33
y
y
−=
4
3
y
=−
The solution set is
4, 1 .
3
−
25.
415
4
x
xx
−=+
()( )
2
15 16 0
1160
xx
xx
−−=
+−=
1 0 or 16 0
1 16
xx
xx
+= − =
=− =
The solution set is
{
}
1,16 .
−
Chapter 7 Rational Expressions
26.
16
23 2
x
xx
−=
+−
Restrictions:
3,2
2
xx
≠− ≠
LCD =
()()
23 2xx
+−
27.
2
24
1
1
24
(1)(1) 1
x
x
xx x
=+
−
=
+− +
Restrictions:
1, 1xx
≠− ≠
LCD =
(1)(1)xx
+−
28.
2
31
11
31
1( 1)( 1)
xx
xxx
=
+−
=
++−
31
1( 1)( 1)
31
(1)(1) (1)(1)
1(1)(1)
3( 1) 1
xxx
xx xx
xxx
x
=
++−
+− =+−
++−
−=
LCD =
1x
−
() ()
()
111
5
11
111
151
11
11511
15 5 11
xx
xx
xx
x
x
+=
−−
−+=−
−−
+−⋅=
+−=
()
3474
37 28 4
725 4
721
x
x
x
x
−+⋅=−
−−=−
−− =−
−=
Section 7.6 Solving Rational Equations
31.
88
4
11
y
yy
=−
++
Restriction: 1y≠−
LCD = 1y+
88
4
11
y
yy
=−
++
32.
22
22
y
yy
=−
−−
Restriction: 2y≠
LCD = 2y−
22
22
2
y
yy
y
=−
−−
=
The proposed solution, 2, is not a solution because
of the restriction 2y≠. Notice that 2 makes two of
the denominators zero in the original equation.
33.
38
3
1xx
+=
−
Restrictions:
1, 0xx
≠≠
LCD =
()
1xx
−
38
3
1
xx
+=
−
3 2 4
2
3
xx
x
==
=
The solution set is
2,4 .
3
34.
24
2
2xx
+=
−
2
3
() () ()
()
2
24
2222
2
24 22 4
xx xx xx
xx
xx xx
−⋅ + −⋅= −
−
+−=−
Chapter 7 Rational Expressions
35.
312
5
44
y
yy
−=
−−
Restriction: 4y≠
The proposed solution, 4, is not a solution because
of the restriction 4y≠.Therefore, this equation has
no solution. The solution set is
{
}
.
36.
10 5
3
22
y
yy
=−
++
Restrictions: 2y≠−
LCD = 2y+
{
}
{
}
37.
11 2
33
x
xx x
−
+=
−−
Restrictions:
0, 3xx
≠≠
3 0 or 1 0
3 1
xx
xx
−= −=
==
The proposed solution 3 is not a solution because of
the restriction
3x
≠
.
The solution set is
{
}
1.
38.
12
11
x
xxx
+=
−−
Restriction:
1, 0xx
≠≠
Section 7.6 Solving Rational Equations
39.
12
3926412
xx
xx x
++=
++ +
To find any restrictions and the LCD, factor the denominators.
()()()
12
332343
xx
xx x
++=
++ +
40.
312
222 1
yy
+=
−−
1.
5
To find any restrictions and the LCD, factor the denominators.
()
222 1
yy
−= −
41.
2
421
55
25
y
yy
y
+=
−+
−
To find any restrictions and the LCD, factor the first denominator.
Chapter 7 Rational Expressions
42.
()()
2
11 22
44 16
11 22
44 44
xx x
xx xx
+=
+− −
+=
+− +−
43.
2
15 6
42 28
xx xx
−=
−+ −−
Factor the last denominator.
()()
15 6
42 42
xx xx
−=
−+ −+
Section 7.6 Solving Rational Equations
44.
2
65 20
32 6
xx xx
−
−=
+− +−
Factor the denominators.
45.
2
223 65
31 23
xx
xxxx
+−
−=
+− +−
Factor the denominators.
()()
223 65
31 31
xx
xx xx
+−
−=
+− +−
Chapter 7 Rational Expressions
46.
2
2
31215
23 6
xx x
xxxx
−+ −
+=
−+ +−
Factor the last denominator.
()()
2
31215
23 32
xx x
xx xx
−+ −
+=
−+ +−
47.
12
21
VP
VP
=
48.
12
21
VP
VP
=