Chapter 7
Rational Expressions
7.1 Check Points
1. a.
728
840
x
x
−
−
Set the denominator equal to 0 and solve for x.
8400
840
5
x
x
x
−=
=
=
The rational expression is undefined for 5.x=
2.
7287(4) 4
21 7 3 3
xxx
xxx
+++
==
⋅
5.
2
949(37)(37)
28 12 4(7 3 )
xxx
xx
−+−
=
−−
1
−
7.1 Concept and Vocabulary Check
1. polynomials
2. 0
3. factoring; common factors
4. 1
5. 1
1.
5
2x
Set the denominator equal to 0 and solve for x.
30
0
x
x
=
=
The rational expression is undefined for
0x=
.
Chapter 7 Rational Expressions
5.
13
520x−
Set the denominator equal to 0 and solve for x.
6.
17
630x−
Set the denominator equal to 0 and solve for x.
6300
630
5
x
x
x
−=
=
=
The rational expression is undefined for
5x=
.
8.
()()
5
79
x
xx
+
+−
Set the denominator equal to 0 and solve for x.
()()
790
xx
+−=
9.
4
(3 17)( 3)
x
xx−+
Set the denominator equal to 0 and solve for x.
10.
()()
8
419 2
x
xx−+
Set the denominator equal to 0 and solve for x.
The rational expression is undefined for x = 19
4 and
x = −2.
11.
2
5
12
x
xx
+
+−
Set the denominator equal to 0 and solve for x.
2
12 0
xx
+− =
()()
2
920 0
450
x
xx
−+ =
−−=
4 0 or 5 0
4 5
xx
xx
−= −=
==
Because the denominator, 5, is not zero for any
value of x, the rational expression is defined for all
real numbers.
x+
Section 7.1 Rational Expressions and Their Simplification
15.
2
3
43
y
yy
+
+−
Set the denominator equal to 0 and solve for x.
2
430
yy
+−=
16.
2
8
62
y
yy
+
−−
Set the denominator equal to 0 and solve for x.
2
620
yy
−−=
17.
2
5
25
y
y
+
−
Set the denominator equal to 0 and solve for x.
2
25 0
y
−=
18.
2
7
49
y
y
+
−
Set the denominator equal to 0 and solve for x.
2
19.
2
5
1x+
The smallest possible value of
2
x is 0, so
2
11x+≥
for all real numbers of x. This means that there is no
Thus, the rational expression is defined for all real
numbers.
21.
2
14 2 7 2 2
77 1
xxxx
x
xx
⋅⋅⋅
===
⋅
25.
28 4
4222
xx
xxx
−−
==
⋅
26.
()
33
39 3
6322
x
xx
xxx
−
−−
==
⋅
29.
()
15 15 5 5
or
393 3 3 3xxx x
−−−
== −
−−− −
()
73
−
Chapter 7 Rational Expressions
32.
()
52
510 5
22
x
x
xx
−
−==
−−
33.
()()
2
551
55 5
25
xx
xx x
x
++
==
+− −
−
37.
()( )
2
111
13 3
23
xx
xx x
xx
++
==
+− −
−−
38.
()()
2
221
23 3
6
xx
xx x
xx
++
==
+− −
−−
39.
()
()()
2
42
48 4
22 2
44
x
x
xx x
xx
−
−==
−− −
−+
43.
()()
()()
2
2
21 3
273 3
21 2 2
252
yy
yy y
yy y
yy
−−
−+ −
==
−− −
−+
44.
()()
()()
2
2
32 2
344 2
3221 21
62
yy
yy y
yy y
yy
−+
+− +
==
−+ +
−−
46. 37
310
x
x
+
+
The numerator and denominator have no common
factor (other than 1), so this rational expression
cannot be simplified.
2
()
()
2
2
21
2
1
xx
x
x
−+
=−
=+
50.
()()
()
()
2
32
2
2
43 4
4312
44
43
4
3
xx x
xxx
xx
xx
x
x
+− +
+−−
=
++
+−
=+
=−
525
5
xx
x
++
=+
53.
()()()
()()
2
2
444
4
44 4
16
xxx
x
xx x
x
−−−
−
==
+− +
−
54.
()()()
2
555
5
xxx
x
+++
+
==
Section 7.1 Rational Expressions and Their Simplification
56.
7
x
x+
; The numerator and denominator have no
common factor (other than 1), so this rational
expression cannot be simplified.
58.
2
5
25
x
x
+
+
; The numerator and denominator are both
prime polynomials. They have no common factor
(other than 1), so this rational expression cannot be
simplified.
61. The numerator and denominator of this rational
expression are additive inverses, so
23 1.
32
x
x
−=−
−
62. The numerator and denominator of this rational
expression are additive inverses, so
54 1.
45
x
x
−=−
−
65.
()()
22 3 23 2
46 2
32 32 32
xx
x
xx x
−−−
−== =−
−− −
67.
()
()
()
2
22 3
46
32
32
23 2
32
x
x
xx
xx
x
xx
−
−=−
−
−−
=−
()
()
35 3
53
33
x
xx
xx
−−
=−
−
==−
()()
()
212
2
12 2
xx
x
xx
+⋅− −
=−
=− + =− −
71.
()()
()()
2
43
12
44
14 3
yy
yy
yy
yy
−+
−− =
−−
−− +
=−
Chapter 7 Rational Expressions
73.
()
()
()
()
2
22
33 3
2
3
1
1
11
1
1
xy
xy x
xxyx y
xy
xy
x
−
−=
−−
⋅− −
=
−
=−
76.
()()
()()
22
22
52
310
32
372
5
3
xyxy
xxyy
xyx y
xxyy
xy
xy
+−
+− =−−
−+
+
=−
78.
()
()
()()
2
3
2
2
224
8
24
28
24
4
xxx
x
xx
xx
xx
x
−++
−=−+
+−
++
=+
80.
()
()()
()()
()()
()
()
2
2
44
16
816 816
144 14
44 4
44
or 144
yy
y
yy yy
yy y
yy y
yy
yy
−+
−=
−+ −+
−⋅ − + − +
==
−− −
++
=
−− −
()()
()()
47 7
47 7
xy y
xx x
+− −
==
++ +
83.
()
()
2
2
32
24 2 1
842
18 12 12 4
xx
xx
xxxx
++
++
=
−−++
()
()
27 339
3
xxxx
x
x
++−+
=+
85.
130
100
x
x−
Section 7.1 Rational Expressions and Their Simplification
This means it costs $520 million to inoculate
80% of the population.
x = 90:
b. Set the denominator equal to 0 and solve for x.
100 0
100
x
x
−=
=
The rational expression is undefined for x = 100.
86. a. x = 20:
()
60,000 20
60,000
100 100 20
1, 200,000 15,000
x
x=
−−
==
b. Set the denominator equal to 0 and solve for x.
100 0
100
x
x
−=
=
87.
1000, 8DA==
1000 8
DA
⋅
=
4000 250
16
==
The correct dosage for a 4-year old is 250
milligrams.
100 100, 000x
+
()
100 500 100, 000
500
150,000 300
500
C+
=
==
the more the equipment is used, the lower the
cost per bicycle.
90. a. x = 100
()
20 100 20,000
Chapter 7 Rational Expressions
The cost per canoe when manufacturing 100
canoes is $220.
c. The cost per canoe decreases as more canoes are
manufactured. One possible reason for this is
that there could be fixed costs for equipment, so
the more the equipment is used, the lower the
cost per canoe.
91.
2
5;3
1
x
yx
x
==
+
represented on the graph by the point
()
3,1.5 .
92. The graph shows that the drug reaches its maximum
concentration after 1 hour. If 1x=,
93. – 97. Answers will vary.
98. makes sense
99. makes sense
100. does not make sense; Explanations will vary.
Sample explanation: 7 is not a factor of 14 .x+
104. true
105. false; Changes to make the statement true will vary.
Therefore,
()()
23
63
22
xx
xx x
xx
+−
−− ==−
++
So
2
6
2
xx
x
−−
+is the desired rational expression.
109. The graphs coincide.
This verifies that the
simplification 315
3, 5,
xx
+=≠−
11
21, 1
xx
xx
−−
=+ ≠
Change the expression on the right from
2
21x− to
()
1
1, 0
xx
xx
xx
xx
−
−=
=− ≠
Change the expression on the right from
2
1 to 1.xx−−
112. Answers will vary.
Section 7.2 Multiplying and Dividing Rational Expressions
115. 25 2
34 20
xy
xy
−=−
+=
Multiply the first equation by 3 and the second
equation by
615 6
xy
−=−
116.
23 6
57 35
⋅=
117.
31 32 6 23 3
42 41 4 22 2
⋅
÷=⋅= = =
⋅
7.2 Check Points
1.
()
()
95
95 945
42 4228
x
xx
xxx
−
−−
⋅= =
+++
3.
2
545(2)(2)
29 45 2 9( 5)
5
xx xxx
xx x x
x
−−−+−
⋅=⋅
−− − −
−
=
1
2x−
(2)(2)xx+−
⋅
1
9( 5)x−
5( 1) 5( 1)
or
7(2 3) 7(2 3)
xx
xx xx
−+ +
=−
−−
5.
437
(3) 71 4
(3)(7)
xxx
xxx
xx
−++
+÷ = ⋅
+−
++
(3)(2) 5
(5)(5) 2
(3)(2)
xx x
xx x
xx
++ +
=⋅
+− +
++
=(5)x+
5
(5)
x
x
+
⋅
−2x+
3
5
x
x
+
=−
Chapter 7 Rational Expressions
7.2 Concept and Vocabulary Check
1. numerators; denominators; PR
QS
7.2 Exercise Set
1.
()
()
45
45 420
39 39927
x
xx
xxx
−
−−
⋅= =
+++
2. 85840
23 36
xx
xx
++
⋅=
−−
6.
75 75 1 1
35 5 7 1
xx
xx
⋅
⋅=⋅ ==
⋅
7.
()
()
45
34 20 3 4
59 27 59 3 9
x
xx x
xx x x
+
−+ −
⋅=⋅ =
+− + −
11.
()()
()()
()
2
2
55 2
25 2
25
310
5
xx x
xx
xxx x
xx
x
+− +
−+
⋅= ⋅
+−
−−
+
=
2
y
=
14.
()
()
()
()
2
33 7 2
921 2
37 2 37
2
3
yy
yy
yyy y
yy
y
+−
+−
⋅= ⋅
+−+
−
=
16.
22
22
31710 432
32216 848
yy yy
yyyy
++ −−
⋅
−− −−
()()
()()
()()
()()
32 5 8 4
32 8 12 4
5
12
yy yy
yy y y
y
y
++ −+
=⋅
+− − +
+
=−
Section 7.2 Multiplying and Dividing Rational Expressions
18.
()
2
3
16 4
yy
−⋅
−
19.
22
22
56 1
23 4
xx x
xx x
−+ −
⋅
−− −
()()
()( )
()()
()()
23 11
13 2 2
1
2
xx xx
xx x x
x
x
−− +−
=⋅
+− + −
−
=+
21.
3
2
82
3
4
xx
x
x
−+
⋅
−
()
()
()()
()
2
2
2242
22 3
24
3
xxx x
xx x
xx
x
−++ +
=⋅
+−
++
=
23.
()
()
32
32
221
44
1
xxx
xx
x
−−+
⋅
−+
−
24.
()
32
32
444
xxx
+++
⋅
25.
22
621
13
xx
xxx
+−
⋅
−+
()
()()
()
()
()
()()
()
()
23 1 1
1131
23 1 1 1
1131
22
xx
xx xx
xx
xx xx
+−
=⋅
+− +
+−−
=⋅
+− +
−
()
()()
()
()
() ()
24 1 1 3
3341
22
or
33
xx
xx xx
xx xx
+−−
=⋅
+− +
−
=−
++
27.
22
22
25 8 20
235 310
yyy
yy yy
−−−
⋅
−− −−
()
()()
()
()()()
23 3
2
3423
2313
2
yy
y
yy y
yy
y
−−
=⋅
−−
−− −
Chapter 7 Rational Expressions
30.
2
22
44
xyxxy
xxy
−+
⋅
−
() ()
()()
44
xy xxy
xxyxy
−+
=⋅ =
+−
32.
22
22
2
2
xy xy
xy xxyy
−+
⋅
+−−
()()
()
()
()()
2
2
2
2
xyxy x y
xy xyxy
xy
xy
+− +
=⋅
++−
+
=+
33.
533
73 75 35
xxx
÷=⋅=
38.
93 94 3322
43 3
322 12
1
xx
xxx x
⋅⋅⋅
÷=⋅= ⋅
⋅⋅
==
40.
()
54 20 5 9
79 7420
59
74 5
xx x
x
x
x
++ +
÷=⋅
+
+
=⋅
+
4
=
42.
()
()
440 4742
67 42 6 40
76
4
6410
7
10
x
xx x
x
x
−
÷=⋅
−− −
−
=⋅
−⋅
=
43.
22
42 42
22
xxxx
xx xx
−+ −−
÷= ⋅
−+
Section 7.2 Multiplying and Dividing Rational Expressions
46.
()
2
2
25
45 7
y
yy y
−
+−÷+
()()()
()()
()( )
2
2
45 7
125
51 7
155
17
5
yy y
y
yy y
yy
yy
y
+− +
=⋅
−
+− +
=⋅
+−
−+
=−
48.
()
()
22
2225
15 5 15 2
25
15 2
3
yyy yy
y
yy
y
y
−−−
÷= ⋅
−
−
=⋅
−
=
51.
22
2
25 10 25
22 45
xxx
xxx
−++
÷
−+−
()()
()
()()
()()
22
2
25 4 5
22 10 25
55 51
21 5 5
5
2
xxx
xxx
xx xx
xxx
x
−+−
=⋅
−++
+− +−
=⋅
−++
−
=
53.
332
22
21
yy yy
yyy y
+−
÷
−−+
()
()
()()
32
232
2
2
21
111
11
yyy y
yyyy
yy yy
yy yy
+−+
=⋅
−−
+−−
=⋅
−−
Chapter 7 Rational Expressions
54.
232
22
312 2
44 2
yyy
yy yy
−−
÷
++ +
y
55.
22
22
5 4 12 35
12 32 3 40
yy y y
yy yy
++ − +
÷
++ +−
56.
22
22
421 1448
328 432
yy y y
yy yy
+− + +
÷
+− +−
()()
()()
()()
()()
22
22
421 432
328 1448
73 84
74 86
3
6
yy yy
yy y y
yy yy
yy yy
y
y
+− +−
=⋅
+− + +
+− +−
=⋅
+− ++
−
=+
58.
2
23
312 12
39
yyy
yy yy
++−
÷
+−
()
()()
343
3
yy y y
++−
=−
22
22
xyxy
+−
3
60.
22
55
7
xyxy
xy
+−
÷−
()()
()()
22
55
7
5
7
5
7
xyxy
xy
xy xy
xyxy
+−
=⋅
−
+−
=⋅
+−
=
Section 7.2 Multiplying and Dividing Rational Expressions
63.
22 2
222
23
212 5 3
xy y x xy y
xx xxyy
−+−
÷
++ + +
()
()()
()()
()()
()
()
22 2
222
2
253
212 3
23
1123
1
xy y x xy y
x x xxyy
yx y x y x y
xx xyxy
yx y
x
−++
=⋅
++ +−
−++
=⋅
++ + −
+
=
+
2
xy
65.
()()
()()()()
()()
22 22
222 2
2412 4 241256
22
918 56 918 4
62 23
2233
63 2 22 3 2 3
yyy y yyyyy
yy
yy yy yy y
yy yy
y yyy
yy y yy y y y
−−− − −−−++
⋅÷=⋅⋅
++
−+ ++ −+ −
−+ ++
− −++
=⋅⋅=⋅=
−− + +− − − −
Chapter 7 Rational Expressions
67.
2232
23
3360 30 310
28 710 25
xx x xx x
xxx x
+− + −
÷⋅
−−+
68.
()()( )
()()
()()
()
()()()
()
()()
()()
()
()()
22 2
22
2
2
2
51 2132 21 1
56221
32 32 5121 21
10 3 1 2
51 3 2 1 51 51 51
32 51 32 32 1 32 1
xx x x x x
xx xx xx
xxxxxx
xx xx
xx x x xx xx x x
xxx xxx
xx
−−++−
−+−−−
÷⋅=÷ ⋅
++−+−
+− −
−+− − − −
=÷ =⋅ =
+−++−
+−
xy yx xy xz xy xz
−− − −−−
71.
33
22 22 33
33 3362
62
99
xy ay xb ab y b xy ay xb ab x a
xa
xa xa yb
++ + + ++ + −
÷= ⋅
−
−−+
72.
33
22 22 33
55 55153
15 3
25 25
xy ay xb ab y b xy ay xb ab x a
xa
xa xa yb
−− + − −− + +
÷= ⋅
+
−−−
Section 7.2 Multiplying and Dividing Rational Expressions
73.
A
lw=⋅
(1)
xx
74.
A
lw=⋅
22
56
36 7 10
56
( 6)( 6) ( 5)( 2)
5
xx
Axxx
xx
xx xx
x
+−
=⋅
−++
+−
=⋅
+− ++
+
=(6)(6)xx+−
6x−
⋅(5)x+
2
(2)
1in.
(6)(2)
x
xx
+
=++
A
76.
1
2
A
bh=⋅⋅
22
2
22
136 10 25
29
25
1292 (5)
2( 5)( 5) 9
xx x
Ax
x
xx
xx x
−+
=⋅ ⋅
−
⋅⋅ −
=⋅ ⋅
+−
Chapter 7 Rational Expressions
81. makes sense
82. does not make sense; Explanations will vary. Sample explanation: You should not look to divide common factors until
after you have changed the division to a multiplication by inverting the second fraction.
87. false; Changes to make the statement true will vary. A sample change is: The quotient of two rational expressions can be
found by inverting the divisor and then multiplying .
88.
? 3 12 3
?22
x
x
−
⋅=
()
34
? 3
?22
x
x
−
⋅=
89. 1 ? 1
23 ? 3x
−÷ =
−
The numerator of the unknown rational expression must contain a factor of 3−. The denominator of the unknown
rational expression must contain a factor of
()
23.x− Therefore, the simplest pair of polynomials that will work are 3−
in the numerator and 23x− in the denominator, to give the rational expression 3.
23x
−
−
Check:
13 1231
2323 23 3 3
x
xx x
−−
−÷=−⋅=
−− −−
Section 7.3 Adding and Subtracting Rational Expressions with the Same Denominator
91. The graph coincides.
This verifies that
3
62
31
xx x x
xx
+⋅=
+.
93. The graphs do not coincide.
()()
()
()
()
22
93 94
44 43
33 4
43
3
xxxx
xx xx
xx x
xx
x
−− −+
÷= ⋅
++ +−
+− +
=⋅
+−
=+
Change the expression on the right from
()()
3 to 3.xx−+
95.
()
233 5
23315
315
18
18
xx
xx
x
x
x
+< −
+< −
−+ <−
−<−
>
()
18,∞
97.
()
2
2
295
295
2950
xx
xx
xx
+=
+=
+−=
1
2
x
=
The solution set is
1
5, .
2
−
98.
7162
99 9 3
−==
99.
23
xx xx+= =
5
=
1
(2)
5
x+
1
2x=+
2.
22
22 2
2
25 10 10 25
25 25 25
xxxx
xx x
−−+
+=
−− −
Chapter 7 Rational Expressions
b.
22
2
2
3 4 11 4 3 4 (11 4)
11 1
34114
1
374
1
(3 4)( 1)
1
34
xxx xx x
xx x
xxx
x
xx
x
xx
x
x
+−+−−
−=
−− −
+− +
=−
−+
=−
−−
=−
=−
5.
22
2
2
421 (1)421
77 7(1)7
421
77
421
7
(3)(7)
7
3
xx x x
xxx x
xx
xx
xx
x
xx
x
x
+−+
+=+⋅
−− −−−
−−
=+
−−
−−
=−
+−
=−
=+
7.3 Concept and Vocabulary Check
1.
PQ
R
+
; numerators; common denominator
2.
PQ
R
−
; numerations; common denominator
3.
1
1
−
−
2.
3811
17 17 17
xx x
+=
3.
893
15 15 15 5
xx x x
+= =
4.
9105
24 24 24 12
xx x x
+= =
5.
35 216 18
12 12 12
xx x
−+ +
+=