Section 7.3 Adding and Subtracting Rational Expressions with the Same Denominator
9.
813217
99 9 3xx xx
+==
10.
41115 5
99 9 3xx xx
+==
14.
97107
44 4
xx x
xx x
++
+=
−− −
15.
()
41891210
6565 65
26 5 2
65
xx x
xx x
x
x
++ +
+=
++ +
+
==
+
18.
2222
22 2
22
33 3
y yyyy yyy
yyyy yy
− + −++
+=
++ +
19.
22 2
2
41314131
55 5
77
5
5
yy yy
yy y
y
y
y
−+ −++
+=
==
()
()()
2
2
22
2
21
22 2
21 2
2
xxx
xx
x
x
xx x
xx
−+ −
=
+−
−
−
== =
+− +
+−
22.
22
22
935
41134113
xx xx
xx xx
+−
+
−− −−
22
2
935
xxxx
++−
=
()()
2
32 3
6
xx x
xx
== =
−+ −
−−
24.
22
xx
−
−=
Chapter 7 Rational Expressions
27.
4343
1
434343
xx
xx x
−
−==
−− −
29.
()
14 7 2
14 7 2
7272 72
14 7 2
72
72
1
72
yy
yy
yy y
yy
y
y
y
−−
−
−=
++ +
−+
=+
+
==
+
31.
()()
31 1
31 1
4242 42
31 1
42
2
xx
xx
xx x
xx
x
x
+− +
++
−=
−− −
+− −
=−
=−
33.
22
33
3161
33
yy
yy
−−
−
34.
22
12 12
yy yy
−
+− +−
()()
22
2
312
12
yyy
yy
+−−
=
+−
22
22
312312
12 12
yyy y
yy yy
+−+ +
==
+− +−
()
34 3
y
+
==
()()
()()
2
2
324
964
38 3 3
3838 38
yy
y
yy y
yy y
+−
=
−
−+ +
==
+− +
Section 7.3 Adding and Subtracting Rational Expressions with the Same Denominator
37.
2
222
62943
299299299
yy y y
yy yy yy
++ −
−−
−+ −+ −+
38.
22
222
32 10 6
310831083108
yyyy
yy yy yy
−+ −
−−
+− +− +−
39.
()
()
1
42 4 2
33 3 13
42
33
2
3
xxx x
xx
x
−
+=+⋅
−− −−−
−
=+
−−
=−
42.
()
()
1
654 65 4
22 2 12
xxx x
xxx x
−
++
+= +⋅
−− −−−
44.
916 2
7337
xx
xx
−−
+
−−
()
()
1
91 62
73 137
xx
xx
−
−−
=+⋅
−−−
()()
2
22
2
2
x
xx
x
x
=−
+−
=−
=+
46.
()
()
22
1
99
33 3 13
xx
xxx x
−
+=+⋅
−− −−−
Chapter 7 Rational Expressions
48.
22
77
16 16
yy
yy
−−
+
−−
()
()
22
1
77
1
16 16
yy
yy
−
−−
=+⋅
−
−−
22 2
77 77
16 16 16
yyyy
yy y
−−+ −−+
=+=
−− −
2
214
16
y
y
−
=
−
50.
()
()
1
10 6 10 6
22 2 12xxx x
−
−=−⋅
−− −−−
10 6 10 6 16
2222xx x x
−+
=−= =
−− − −
52.
()
()
1
11 5 11 5
77717xxx x
−
−=−⋅
+−− + −−−
()
44 4 14
yyy y
− − −−−
44
44 4
yy
yy y
−+
=−=
−− −
55.
()
()
1
3253 25
77 7 17
325
77
xx x x
xxx x
xx
xx
−
−−− −
−=−⋅
−− −−−
−−+
=−
−−
56.
()
()
1
4384 38
99 9 19
xx x x
xxx x
−
−−− −
−=−⋅
−− −−−
()( )
438
438
99 9
xx
xx
xx x
−−−+
−−+
=− =
−− −
43843824
xx xx x
−+ − −+ − −
()( )
22
25 25
22
xx
xx
−−
−−−+
=
Section 7.3 Adding and Subtracting Rational Expressions with the Same Denominator
59.
()
()
1
1
1
xyx y
xy yx xy yx
xy
xy xy
xy
xy
−
+=+⋅
−−−−−
−
=+
−−
−
==
−
61.
22 22
22xy
xy yx
+
−−
()
()
22 22
1
22
1
xy
xy yx
−
=+⋅
−
−−
62.
()
()
22 22 2 2 22
1
222 2
1
yx y x
xy yx xy yx
−
+=+⋅
−
−−− −
2
22
2194
67 67
xx
xx xx
−−+
=+
+− +−
()()
2
2
2
2
219 4
67
421
67
73 3
xx
xx
xx
xx
xx x
−− +
=
+−
+−
=
+−
+− −
()()
22
30 30
51
56 6
xx xx
x
xx x
==
−− −−
+
==
+− −
22 2
()()
2
4943
16 48 27
4
bb
bb
b
==
−−
−+
=−
Chapter 7 Rational Expressions
67.
2222222
555 5 5 555
yyyyyyy
yyy y y yyy
−+−
−+ =− =−=
−−− − − −−−
69.
ba
ac ad bc bd ac ad bc bd
−
+−− +−−
()()
()()
()
()
()()()()
111
or
1
ba ba
ac ad bc bd a c d b c d
ba ba ab
cdab cdab cdab cd cd
−−
==
+−− +− +
−
−−−−
==⋅= =−
+− − +−−+− + +
71.
()()
()( )
()()
()( )
()()
()( )
()( )()( )()( )
22 2
32 23 51 65645
1 4 14 14 1 4 14 14
yy yy yy yy y y y y
yyyyyyyyyyyy
−+ ++ +− −− + + + −
−−=−−
+− +− +− +− +− +−
72.
(
)
(
)
()()
(
)
(
)
()()
(
)
(
)
()()()()()()()()
22 2
12 1 2 1 52 1 21 22115
23 2332 23 231312
yy yy yy yy yy y y
yy yy yyyy yy y y
+− +− ++ +− +− + +
+− =+−
−− −− −− −− −−−−⋅−−
Section 7.3 Adding and Subtracting Rational Expressions with the Same Denominator
b.
100 ; 5, 6
WWL
L==
100 100 5 83.3
6
W
L
⋅
=≈
Since this value is over 80, the skull is round.
74. a.
22
30 50
tt
+−
−
b.
1:t=
22
80 80
411 411
tt
=
++ +⋅+
75.
22
510 5
22
33
PLW
x
xx
=+
+
=+
++
8
4
x
==
+
The perimeter is 8 inches.
77. – 80. Answers will vary.
81. does not make sense; Explanations will vary.
Sample explanation:
31 243
44 4
xx x++ +
+=
and
43x+
is not divisible by 4.
denominator by –1.
85. false; Changes to make the statement true will vary.
A sample change is: You do not add the common
88. true
Chapter 7 Rational Expressions
89.
22 2
31 2 7 2
56 56 1
xxx
xx xx x
−−+
−÷
+− +− −
()( )
22
31 2 7 2
56 1
xx x
xx x
−− − +
=÷
+− −
90.
2
222
344 109 5
372372 4
xx x x
xx xx x
−+ + −
−÷
++ ++ −
()
()
2
22
344109 5
372 4
xx x x
xx x
−+− + −
=÷
++ −
91.
2?41
33 3
xx
xx x
+
+=
++ +
The sum of numerators on the left side must be
92.
3?617
22 2
xx
xx x
−
−=
++ +
The difference of the numerators on the left side
must be
617,
x−
so the missing expression is
20 6.
x−
93.
()
()
()
22 2
1
6?13
212 2
1?
613
22 2
xxx
xxx
xx x
+=
−− −
−
+⋅ =
−−− −
−
+=
−− −
Since
6713,
+=
the opposite of the missing
must be
()()
2
43 12
44
aa aa
aa
−+ −−
=
−−
.
Since
22
12 12,
aa aa−+ = −−
the missing
Section 7.4 Adding and Subtracting Rational Expressions with Different Denominators
95.
()
()
3?71
55 5
1
3?71
515 5
xx
xxx
xx
xxx
+
+=
−− −
−+
+⋅ =
−−− −
96. The graphs coincide.
This verifies that
36 3.
22
xx
x
+−=+
97. The graphs do not coincide.
24359
22
xx x
xx
++ +
−
++
98. The graphs do not coincide.
()()
22
13 3 16
44 4
44
4
4
xx
xx x
xx
x
x
−−
−=
++ +
+−
=+
=−
100.
()()
()
()()
422
2
81 1 9 1 9 1
913131
xxx
xxx
−= + −
=++−
2
2
375
3
x
=−−
+
102.
12347
23 66 6
+=+=
103.
15 3 20 17
8 6 24 24 24
−= − =−
1.
2
37
and 15
10 x
x
List the factors for each denominator.
22
10 2 5
15 3 5
xx
xx
=⋅
=⋅
LCD =
22
235 30xx⋅⋅⋅ =
Chapter 7 Rational Expressions
3.
22
911
and
728 816xxxx+++
List the factors for each denominator.
2
22
7287(4)
816( 4)
xxxx
xx x
+= +
++=+
LCD =
2
7( 4)xx+
5.
24
+
33xx+−
LCD =
()()
33xx+−
()()()()
()()
()()
24 32 34
++
33 3333
26 412
+
33 33
26412
33
66
33
xx
xx xxxx
xx
xx xx
xx
xx
x
xx
−+
=⋅ ⋅
+− −++−
−+
=+− +−
−+ +
=+−
+
=+−
7.
2
55
5 25 ( 5) 5( 5)
5
yy
yyy y
yy
−= −
−−−
−
LCD =
5( 5)yy−
2
55
5 25 ( 5) 5( 5)
5
55
yy
yyy y
yy
yy
−= −
−−−
−
=⋅ −⋅
5( 5)yy
−
1
5
5
y
y
+
=−
8.
2
43 4 3
5(5)(5)5
25
xx
xx x x
x
+= +
−+−−
−
LCD =
(5)(5)xx+−
2
43 4 3
5(5)(5)5
25
41(5)3
(5)(5) 1(5)5
41(5)3
( 5)( 5) ( 5)( 5)
xx
xx x x
x
xx
xx x x
xx
xx xx
+= +
−+−−
−
−+
=+⋅
+−−+ −
−+⋅
=+
+− +−
7.4 Exercise Set
1.
2
713
and 24
15 x
x
22
3
15 3 5
24 2 3
xx
xx
=⋅
=⋅
LCD =
32 2
235 120xx⋅⋅ =
4.
25
711
and
15 24xx
22
535
15 3 5
24 2 3
xx
xx
=⋅
=⋅
LCD =
35 5
235 120xx⋅⋅⋅ =
()()
7.
()
510
and
72yy+
LCD =
()
72yy+
9.
2
17 18
and
416
xx
+−
()
()()
2
41 4
16 4 4
xx
xxx
+= +
−=+ −
LCD =
()()
44xx+−
34
12.
()
2
14 12
and 7
49 yy
y−
−
()()
()()
2
49 7 7
77
yyy
yy yy
−=+ −
−= −
LCD =
()()
77yy y+−
14.
22
25 10 25
yyy−−+
()()
()()
2
2
25 5 5
10 25 5 5
yyy
yy yy
−=+ −
−+=− −
LCD =
()()()
555yyy+−−
Chapter 7 Rational Expressions
16.
22
7 and
56 45
x
xx xx−− −−
()()
()( )
2
2
56 6 1
45 1 5
xx x x
xx x x
−−=− +
−−=+ −
LCD =
()()()
615xxx−+−
xxx
18.
2
48
xx
+
LCD =
2
x
222
48 4 8 48xx
xxx
xxx
+
+=⋅+=
20.
57
68xx
+
LCD =
24x
5 7 54 73
68 6483
20 21 41
24 24 24
xx x x
xx x
⋅⋅
+= +
⋅⋅
=+=
22.
2
10 3
5
xx
+
LCD =
2
5x
222
10 3 10 5 3 50 3
5
555
xx
xxx
xxx
⋅+
+= +=
⋅
23.
1
6x
+
LCD = x
16 16 161
61
xxx
xxxxxx
+
+=⋅+= +=
1
25.
29
x+
LCD = x
2292929
91
xxx
xxxxxx
+
+= +⋅ = + =
26.
74
()
()
2
121 2
636 32
31
12 4 3 3
66 6 6
1
2
x
xx x
x
xx x
x
+
−+ −
+=+ ⋅
+
−+ +
=+ = =
+
=
Section 7.4 Adding and Subtracting Rational Expressions with Different Denominators
29.
43
5xx
+−
LCD =
5xx−
30.
34
6xx
+−
LCD =
()
6xx−
31.
23
12xx
+
−+
LCD =
()( )
12xx−+
23
12xx
+
−+
32.
34
23xx
+
−+
LCD =
23xx−+
33.
23
54yy
+
+
LCD =
()
45yy+
34.
32
13yy
+
+
LCD =
()
31yy+
32
13
yy
+
+
Chapter 7 Rational Expressions
35.
1
7
x
x−
+
LCD =
7x+
36.
1
6
x
x−
+
LCD =
6x+
37.
74
55xx
−
+−
LCD =
()()
55xx+−
38.
82
66xx
−
+−
()()
()()
66
660
66
xx
x
xx
=+−
−
=+−
()()
()()
()
()()
2
44 4
4
2
44 44
xx
xx x
xx
x
xx xx
=+
+− −
+
=+
+− +−
Section 7.4 Adding and Subtracting Rational Expressions with Different Denominators
40.
2
4
5
25
xx
x
x
++
−
()()
()
2
25 5 5
51 5
xxx
xx
−=+ −
+= +
LCD =
()()
55xx+−
41.
2
54
3
9
y
y
y
−+
−
LCD =
()()
33yy+−
2
54
3
9
y
y
y
−+
−
42.
2
85
4
16
y
y
y
−+
−
LCD =
()()
44yy+−
2
85
4
16
y
y
y
−+
−
()()
111xxx
−−−
LCD =
()()
11xx−−
()()
73
111xxx
−
−−−
()
()()
()()
71 3
11 11
x
xx xx
−
=−
−− −−
()()
()()
5152
33
513
33
x
xx
x
xx
+−
=++
+
=++
Chapter 7 Rational Expressions
45.
39
420630
yy
yy
+
−−
()
()
4204 5
6306 5
yy
yy
−= −
−= −
46.
43
51010 20
yy
yy
+
−−
()
()
5105 2
10 20 10 2
yy
yy
−= −
−= −
47.
4
4
yy
yy
+−+
LCD =
()
4yy+
4
4
yy
yy
+−+
()
()()
()
()()
()
2
55
55
55
5
yy
yy
yy yy
yy y
yy
−−
⋅
=−
−−
−− −
=−
Section 7.4 Adding and Subtracting Rational Expressions with Different Denominators
49.
2
29 2
3
712
x
x
xx
+−−
−+
()()
()
2
712 3 4
31 3
xx x x
xx
−+=− −
−= −
50.
2
37 3
3
56
x
x
xx
+−−
−+
()()
()
2
56 3 2
31 3
xx x x
xx
−+=− −
−= −
LCD =
()()
32xx−−
51.
()
22
34
11
xx
+
−+
()()
2
2
111
xxx
−= + −
()()()
111
xxx
=++−
52.
()
22
62
42
xx
+
−+
()()
2
2
422
xxx
−= + −
Chapter 7 Rational Expressions
53.
22
32
310 6
xx
xx xx
−
+− +−
()()
()()
2
2
310 2 5
632
xx x x
xx x x
+−=− +
+−= + −
54.
22
224 76
xx
xx xx
−
−− −+
()()
()()
2
2
224 6 4
76 6 1
xx x x
xx x x
−−=− +
−+=− −
LCD =
()()()
641xxx−+−
55.
22
4
21 54
y
yy yy
+
++ ++
()()
()()
2
2
21 1 1
54 4 1
yy y y
yy y y
++=+ +
++=+ +
56.
22
4
56 6
y
yy yy
+
++ −−
()()
()()
2
2
56 2 3
632
yy y y
yy y y
++=+ +
−−= − +
LCD =
()()()
233yyy++−
Section 7.4 Adding and Subtracting Rational Expressions with Different Denominators
57.
53
35
xx
xx
−+
+
+−
LCD =
()()
35xx+−
53
35
xx
xx
−+
+
+−
58.
74
47
xx
xx
−+
+
+−
LCD =
()()
47xx+−
74
47
xx
xx
−+
+
+−
59.
2
53
22
22 y
yy
−−
−
()
()
2
222 1
222 1
yyyy
yy
−= −
−= −
LCD =
()
21yy−
60.
2
72
55
55 y
yy
−−
−
()
()
2
555 1
555 1
yyyy
yy
−= −
−= −
()()
(
)
(
)
()()
()()()
()()
2
13
43 1 43
333 33
9
43 1 3
33
xx
xx x
xxx xx
x
xxx
xx
++
++ +
−= −
−+− +−
−
+−+ +
=+−
LCD =
()()
66xx+−
2
2165
636
xx
xx
−−
−
+−
()()
()()
21 65
666
21 6 65
xx
xxx
xx x
−−
=−
++−
−− −
=−
Chapter 7 Rational Expressions
63.
2
2
39 7
2
310
yy
y
yy
−−
−−
+−
()()
()
2
310 2 5
21 2
yy y y
yy
+−=− +
−= −
LCD =
()()
25yy−+
64.
2
2
64
6
918
yy
y
yy
−−
−+
++
()()
()
2
918 3 6
61 6
yy y y
yy
+−=+ +
+= +
LCD =
()()
36yy++
()()
()()
()
()()
2
22
22
643
36
612
36
612
yyy
yy
yyy
yy
yyy
−− − +
=++
−− −−
=++
−− ++
()
()
75
11
7555
751
5
7351
5
x
xxx
x
x
x
x
−
+= +
−−−
−+
=−
−+
=−