Section 7.4 Adding and Subtracting Rational Expressions with Different Denominators
68.
4
75
y
y
−+
LCD = 5y+
69.
2
93
3
6
xx
x
xx
++−
−−
()()
()
2
632
313
xx x x
xx
−−= − +
−=− −
LCD =
()()
32xx−+
70.
2
2
95
3
23
xx
x
xx
++−
−−
()()
2
23 3 1xx x x−−=− +
()()
()
()()
()
()()
2
2
2
51
9
31 31
95 1
31
x
xx
xx xx
xxx
xx
−+
+
=+
−+ −+
+− +
=−+
()()( )
()
()()( )
()()()
()()( )
()()( )
()()( )
2
2
11 2
22
11 2
3122
11 2
4324
11 2
21
11 2
xxx
x
xxx
xx x
xxx
xx x
xxx
xx
xxx
=+−+
+
−+−+
++−+
=+−+
++−−
=+−+
+−
=+−+
Chapter 7 Rational Expressions
72.
2
4
210
10 25
xx
x
xx
−
−−
−+
()()
()
2
10 25 5 5
2102 5
xx xx
xx
−+=− −
−= −
()()
()()
()()
()()
245
25 5
xx x
xx
−− −
=−−
73.
2
35
15
5
yy
y
y
+−
−
LCD =
2
15y
2
35
15
5
yy
y
y
+−
−
74.
2
72
12
3
yy
y
y
−−
−
LCD =
2
12y
2
2
2
72
12
628
yy
y
yy
−−
=
−+ −
=
LCD =
()()
32 2xx+−
2
3
36
4
xx
xx
++
+−
()()()
()
3
322 2
1
3
xx
xxx
xx
+
=+
++−
−
+
Section 7.4 Adding and Subtracting Rational Expressions with Different Denominators
76.
2
7
412
9
xx
xx
++
+−
()
()()
2
4124 3
933
xx
xxx
+= +
−=+ −
()
()
()
()()
1
7
43 13 3
xx
xxx
−
+
=+⋅
+−+−
77.
22
2
1
yy
yyy
+
−−
()()
()
2
2
111
1
yyy
yy y y
−= + −
−= −
Note that
()
11 1yy−−=−
LCD =
()()
11yy y+−
()()
()
()()
()
222
21
11 11
21 22
yy
yy
yy y yy y
yyy yyy
−+
⋅
=+
+− +−
−+ −−
==
78.
22
1
yyy
+
−−
()()
2
111
yyy
−= + −
()()
() ()
()()()
11 11
5
11 1
yy yy
yy
yy yy
=+⋅
+− − −
−
=+
+− −
()()
()
()()
()
222
51
11 11
51 55
yy
yy
yy y yy y
yyy yyy
−+
⋅
=+
+− +−
−+ −−
==
Chapter 7 Rational Expressions
79.
11xy
xy
−+
+
LCD =
xy
80.
22xy
yx
+−
+
LCD =
xy
81.
22
32x
yx
xy
−−
−
()()
22
xy xyxy−=+ −
Note that
()
1yx xy−=− −
LCD =
()()
xyxy+−
Section 7.4 Adding and Subtracting Rational Expressions with Different Denominators
82.
22
73x
yx
xy
−−
−
()()
22
xy xyxy−=+ −
Note that
()
1yx xy−=− −
LCD =
xyxy+−
83.
2
633
22
4
xxx
xx
x
++−
−+
+−
−
LCD =
()()
22xx+−
()()
2
633 6 33
22 22 22
4
xxx x xx
xx xx xx
x
++− + +−
−+= −+
+− +− +−
−
84.
2
822
33
9
xxx
xx
x
++−
−+
+−
−
LCD =
()()
33xx+−
()()
2
822 8 22
33 33 33
9
xxx x xx
xx xx xx
x
++− + +−
−+= −+
+− +− +−
−
Chapter 7 Rational Expressions
85.
22 2
54 3
25 11 30 30xxxxx
+−
−−+−−
()()
2
25 5 5
xxx
−=+ −
86.
22 2
32 5
49 15 56 56xxxxx
+−
−−+−−
()()
()()
2
2
49 7 7
15 56 7 8
xxx
xx xx
−=+ −
−+=− −
87.
332
6
27 3 9
xx
xxxx
+−
−++
()
()
32
27 3 3 9
xxxx
−=− ++
Section 7.4 Adding and Subtracting Rational Expressions with Different Denominators
88.
332
8
824
xx
xxxx
+−
−++
()
()
()
32
32 2
8224
24 24
xxxx
xxxxxx
−= − + +
++= ++
89.
2
93 1
32
6
yyy
yy
yy
+−
++
−+
−−
()()
()
()
2
632
313
21 2
yy y y
yy
yy
−−= − +
−=− −
+= +
Chapter 7 Rational Expressions
90.
2
72 2 1
43
12
yyy
yy
yy
−+
++
−+
−−
()()
()
2
12 4 3
414
yy y y
yy
−− = − +
−=− −
91.
222222
352
43 23 9xxyyxxyyxy
−+
++ −− −
()( )
()()
22
22
22
43 3
23 3
933
xxyyxyxy
xxyyxyxy
xyxyxy
++ =+ +
−− =− +
−=+ −
()( )( )
Section 7.4 Adding and Subtracting Rational Expressions with Different Denominators
92.
222222
574
32 2 4xxyyxxyyxy
−+
++ −− −
()( )
()( )
22
22
32 2
22
xxyyxyxy
xxyy xyxy
++ =+ +
−− =+ −
93. Young’s Rule:
12
DA
C
A
=+
8;A=
88 2
812 20 5
DDD
C⋅
===
+
3;A=
94. Young’s Rule:
12
DA
C
A
=+
10;A=
10 10 5
10 12 22 11
DDD
C⋅
===
+
Chapter 7 Rational Expressions
95. Young’s Rule:
12
DA
C
A
=+
Cowling’s Rule:
()
1
DA
C+
=
96. Cowling’s Rule:
()
1
24
DA
C+
=
12 :A=
()
12 1 13
24 24
DD
C+
==
10 :A=
and a 10-year-old child is
12.
D
This means that a
12-year-old should be given
1
12
of the adult dosage
more than a 10-year-old.
97. No, because the graphs cross, neither formula gives
a consistently smaller dosage.
101.
22PLW=+
22
34
22
xx
xx
xx
=+
+−
()
()()
()
()()
()()
22
56
26 25
56 56
212210
56
xx
xx xx
xx xx
xxxx
xx
=+
++
++
=+
++ ++
+++
=++
105. Explanations will vary. The right side of the
equation should be charged from
3
5x+
to
52
5
x
x
+
.
106. Explanations will vary. The right side of the
equation should be changed from
1
7x+
to
17.
x
x
+
Section 7.4 Adding and Subtracting Rational Expressions with Different Denominators
111. does not make sense; Explanations will vary.
Sample explanation: It is acceptable to leave the
numerator in factored form.
114. false; Changes to make the statement true will vary.
A sample change is:
22
11 1
.
1
xxx
xxxx
+
+=+ =
115. true
116.
22
22
54 6 2
1
23 23
yy yy
y
yyyy
++ +−
⋅−
−
+− +−
()()()
()()()
32 2
32
368246
311
10 2
311
yyy yy
yyy
yy y
yyy
+−−−−+
=+−−
+− −
=+−−
117.
11
h
xh x
−÷
+
() ()
11
h
xx h h xx h
−−
=⋅=
++
118.
2 ?
1?x−
−
()
2
2
231
1
xx
xx
+−
=
−
If the first rational expression is multiplied by
2
2
,
x
x
22
()()
428
21 21
yx
xx xx
+
−=
−+ −+
Then,
()( )
()()()()
41 2 28
21 21
xyx x
xx xx
+− − +
−
−+ −+
So 44 2 28,xyxyx+− + = + which implies that
42xyx x−= and 42 8y+=
. Both of these
equations give 2.y= Thus, the missing rational
expression is
2
1
x+
.
Chapter 7 Rational Expressions
120.
()()
3527
xx+−
2
2
6211035
61135
xxx
xx
=−+−
=−−
122. First find the slope
()
()
04
41
134
m−−
===
−−
123. a.
121523 5 6 11
3 5 3 5 5 3 15 15 15
⋅⋅
+= + = + =
⋅⋅
b.
21 2315 6 5 1
5 3 5 3 3 5 15 15 15
⋅⋅
−= − =−=
⋅⋅
b.
111 1
1
yx
xy x y xy xy
xy
+
÷+ =÷
=⋅
()()
2280
420
xx
xx
−−=
−+=
4 0 or 2 0
4 2
xx
xx
−= +=
==−
The rational expression is undefined for
4x=
and
2.x=−
2.
()()
()()
2
2
31 2
372
3121
61
xx
xx
xx
xx
−−
−+
=−+
+−
()
()()
33 3
32 2
y
yy y
−− −
==
−− −
4.
()
()
2
32
43 3
823
16 24 2
812 423
ww
ww
w
ww ww
−
−==
−−
Mid-Chapter Check Point
6.
2
28
24 4
x
xx
+⋅
−−
8.
()()
()()
()()
()()
22
22
22
22
2134
273 6
21 6
273 34
21 1 3 2
21 3 4 1
2
4
xx x x
xx xx
xx xx
xx xx
xx x x
xx x x
x
x
+− − −
÷
−+ −−
+− −−
=⋅
−+ −−
−+ − +
=⋅
−− − +
+
=−
9.
22
11
23 56
xx xx
+
+− ++
10.
17 8
55
x
xx
+
+
−−
Chapter 7 Rational Expressions
11.
()()
()
()()
()()
22
22
2121 4 3
41 712
33 2 1 4
93 2 74
yy yy
yyy
yy y y
yy y y
+− −−
−−+
⋅= ⋅
−+−
−−−
12.
2
12
yy
yy
−
++
LCD =
()( )
12yy++
14.
()()
22
2
2
25
943
933
43 3 1
z
zzz
zzz
zz z z
−
−++
−= + −
++=+ +
15.
()
2
25
31 31
z
zz
++
−−
Mid-Chapter Check Point
16.
()()
()
2
2
83
7
421
421 7 3
71 7
x
xx
xx x x
xx
++
+−
+−=+ −
+= +
17.
4
22
27 3
939
xxx
xxx
−+
⋅
−++
18.
()()
()()
22
2
2
12
243
221
43 3 1
xx
xx x x
xx x x
xx x x
−+
−
−− + +
−−= − +
++=+ +
LCD =
()()()
213xxx−++
22
12
243
xx
xx x x
−+
−
−− + +
19.
222
22
2
2
55
255
xxyyxxy
xy x y
xxyyxy
xy xxy
−+ −
÷
++
−+ +
=⋅
+−
2
5118
54 20
5118
xx
xxxx
xx
−
+−
+− +−
−
()()
()()
()()
2
34
12
54 54
3
5
xx
xx
xx xx
x
x
+−
−−
==
+− +−
+
=+
Chapter 7 Rational Expressions
7.5 Check Points
1.
12
43
21
34
+
−
Add to get a single rational expression in the
numerator.
2.
1
2
1
2
x
x
−
+
Subtract to get a single rational expression in the
numerator.
3.
11
1
xy
xy
−
Subtract to get a single rational expression in the
Invert and multiply.
11
yx
yxxy
xy yx
xy
xy
−
−
=⋅=−
12
+
1
2
x
+
Multiply the numerator and the denominator by the
LCD of x.
1
11
2
22
21
11
121
xxx x
x
xx
x
−
−⋅−⋅
−
== =
+
11
11
xy xy yx
xy yx
xy xy
⋅− ⋅ −
===−
⋅
Section 7.5 Complex Rational Expressions
7.5 Exercise Set
1.
11
24
11
23
+
+
Add to get a single rational expression in the
numerator.
2.
11
34
11
36
+
+
Add to get a single rational expression in the
27
27 10 9 3 2 5 18
5
69 569532323
10
⋅⋅⋅
==⋅= =
⋅⋅
4.
3538
184 32
5555
1817
57 35
24444
++
===⋅=
−−
23
60 60
34
24 20 4 4
40 45 5 5
=
⋅− ⋅
−
===−
−−
11
−
Chapter 7 Rational Expressions
7.
334
444
334
444
x
x
x
x
−−
=
++
8.
2
3
2
3
x
x
−
+
LCD = 3
10.
38 383
8
777
1
83 83
77
xx
xxx x
xx
xxx x
xxx
xx x
+
++
==
−
−−
++
=⋅=
−−
13.
13 23
22
13 43
y
yy
y
−
−
=+
+
14.
4
12
3
y
y
−
+
LCD = 12 y
11
5
x
+
LCD = 5x
5
555
5
11 11
5
55
x
xxx
x
x
xx
⋅−
−
=
+⋅+
Section 7.5 Complex Rational Expressions
17.
()()
2
22
2
2
2
11
1
11
1
1
1
1
11
1
x
xx
x
xx
xx
xx
xx
xx x
x
x
+
+
=
−
−
+
=⋅
−
+
=⋅
+−
=−
19.
11
7
7
7
y
y
−
−
LCD = 7y
11
11 77
yy
−
−
20.
()
11 9 9
9999
999
999
99 9 9
99 9 1 9
11
yy
yyy y
yyy
yy
yyy y
yy
−
−−
==
−−−
−−
=⋅=⋅
−−−
−
==−
21.
22
22
xy
xxy y xy
yy
+
+++
==⋅=
23.
11
xy
xy
+
LCD =
xy
11
11 xy xy yx
xy
+
+
+
==
Chapter 7 Rational Expressions
26.
11
11
xy
+
−
27.
2
12
21
yy
y
+
+
LCD = 2
y
28.
222 2
2
13 3 3
333
1
31
3
yy
yyyy y
yy
yyyy
yy
yy
y
+
++
==
+
++
+
=⋅=
+
30.
2222
82 82 82
10 6 10 6 10 6
xx
x
xxxx
xx
−
−−
==
−
−−
31.
2
6
2
9
1
y
y
+
−
LCD = 2
y
LCD = 2
y
()
2
2
2
2
2
12
3312
16
16
1
34 3
yyyy
y
y
y
yy y
+
+
=
−
−
+
==
22
xx
++