Chapter 6
Rational Expressions and Equations
Exercise Set 6.1
common.
2. −8y=0
4. y+6=0
6. 4x−15=0
8. p2−7p+10=0
10. 49 −x2=0
12. Since the denominator is the constant 14, there are no
numbers for which the expression is not defined.
22. 4x2
20x=4x
4x·x
5=x
5
28. 4y2−2y
5y2−5y=2y/(2y−1)
5y/(y−1) =2(2y−1)
5(y−1)
34. x2+8x+ 16
x2−16 =(x+ 4)✏(x+ 4)
(x+ 4)✏(x−4) =x+4
x−4
40. 8x2−32
4x2−16 =8(x2−4)✏
4(x2−4)✏=2
3/(a−4)(a+ 1)✏
−x+y=y−x
y−x=1
52. 5a−15
60. 3x2y
x
y
7
68. x4−1
70. (t−2)3
(t−1)3·t2−2t+1
t2−4t+4 =
72. 2t2−98
4t2−4·8t+8
16t−112 =
74.
78. 10x2+80x+ 70 = 10(x2+8x+ 7) = 10(x+ 1)(x+7)
=x
Exercise Set 6.2
RC2. (e)
6. x2−3
x2
10 ÷3
2=3
10 ·2
3=3/ ·2/ ·1
5·2/ ·3/ =1
5
14. x2
y÷x3
y3=x2
y·y3
x3=x2
y/·x2
✧·x=y2
x
18. 4y−8
y+2 ÷y−2
y2−4=4y−8
y+2 ·y2−4
y−2=
20. a
a−b÷b
a−b=a
a−b·a−b
b=a(a−b)✏
(a−b)✏b=a
b
4/ ·2/(−3+x)✏·1=3
30. a−b
(y+5)✏·2/ ·2·y/·y
2/ ·y/(y+5)✏(y−5) =2y
y−5
34. x2+x−20
Exercise Set 6.3 135
36. 5t2+5t−30
10t+30 ÷2t2−8
6t2+36t+54 =
5t2+5t−30
2(2h−4)(h)=35
40. (3p2−6pq +7q2)−(5p2−10pq +11q2)=
42. (5x6y−4)3= 125x18y−12 =125x18
y12
46. 3x+3y+3
9x÷x2+2xy +y2−1
x4+x2
48. The volume Vof a rectangular solid is given by the formula
V=l·w·h, where l= the length, w= the width, and h=
Exercise Set 6.3
2. 10 = 2 ·5
4. 12 = 2 ·2·3
6. 8=2·2·2
36 = 2 ·2·3·3
8. 4=2·2
5=5
60 +2
25
=7
2·2·3·5+2
5·5
136 Chapter 6: Rational Expressions and Equations
16. 2
15 +5
9+3
20
20. p3q=p·p·p·q
p2q=p·p·q
24. y=y
26. x2−4=(x+ 2)(x−2)
28. m4−m2=m·m(m+ 1)(m−1)
m3−m2=m·m(m−1)
34. 9−4x2=(3+2x)(3 −2x)
3+2x=3+2x
38. x5−4x3=x·x·x(x+ 2)(x−2)
44. 12ab =2·2·3·a·b
24x3=2·2·2·3·x·x·x
48. The LCM of a5and a15 is a15; the GCF is a5.
Exercise Set 6.4
RC2. The LCD is 2(x+ 3). We have
4. x2+7x
x2−5x+x2−4x
x2−5x=2x2+3x
x2−5x=x/(2x+3)
x/(x−5) =2x+3
x−5
10. LCD = 72x
14. LCD = 7c2d3
16. LCD = c2d2
2c−d
=2cd −d2+c2+cd
c2d2
=c2+3cd −d2
c2d2
18. LCD = (y−1)(y+1)
4
5y+7
y−2
22. 4x
x2−25 +x
x+5
24. t
26. LCD = (y+3)
2
=8+5(y+3)
2
28. 9x
6x−30 +3x
4x−20
=9x
2·3(x−5) +3x
2·2(x−5)
30. a
a−3+a−3
a
32. a
a2−2a+1+1
a2−5a+4
34. 3x
2y−3+2x
3y−2
36. 3x+2
3x+6+x−2
x2−4
38. 5
−3+11
3=−5
3+11
3=6
3=2
44. t2
t−2+4
2−t=t2
t−2+4
2−t·−1
−1
46. a−3
a2−25 +a−3
25 −a2=a−3
a2−25 +a−3
25 −a2·−1
−1
50. 3(x−2)
2x−3+5(2x+1)
2x−3+3(x+1)
3−2x
52. 5(x−2)
3x−4+2(x−3)
4−3x+3(5x+1)
4−3x
=2(x+5)
(2x−3)(x−1) +3x+4
(2x−3)(1 −x)·−1
−1+
56. a−2
3−a+4−a2
a2−9
=(−a+ 2)(a+3)+4−a2
(a+ 3)(a−3)
y
2yx 10 0
Exercise Set 6.5 139
58. x+2
x−7+3−x
49 −x2
60. 10
a2−a−6+3a
a2+4a+4
62. (4y3−5y2+7y−24) −(−9y3+9y2−5y+ 49)
=4y3−5y2+7y−24+9y3−9y2+5y−49
=13y3−14y2+12y−73
68.
72. P=2
3
x+4+2
2
x−5
y2−9+4y
(y−3)2+6
3−y·−1
−1
=−2
(y+ 3)(y−3)2
Exercise Set 6.5
RC2. 7
4. t2
t+5−25
t+5 =t2−25
t+5 =(t+5)✏(t−5)
(t+5)✏·1=t−5
8. y+3
2−y−4
4
140 Chapter 6: Rational Expressions and Equations
10. a−1
4a−2a+3
a
12. 5x+3y
2x2y−3x+4y
xy2
14. 3t
t−1−8t
t+1
16. 11
x2−4−8
x+2
18. 3
12 + x−x2−2
x2−9
20. 3x−2
4x−3x+1
6x
22. a
a−b−a
a+b
24. 5
9−7
−9=5
9−7
−9·−1
−1=5
9−−7
9=5−(−7)
9=
30. t2
t−2−4
2−t=t2
t−2−4
2−t·−1
−1=t2
t−2−−4
t−2=
2(x−8)
(x+ 4)(x−4)
Exercise Set 6.5 141
36. 8x
16 −x2−5
x−4
=8x
(4 + x)(4 −x)−5
x−4
38. 4
5a2−5a−2
5a−5
40. a
a2+11a+30−5
a2+9+20
=a(a+4)−5(a+6)
(a+ 5)(a+ 6)(a+4)
42. a−2b
b−a−3a−3b
a−b+2a−b
a−b
44. x−3y
2(y−x)+x+y
2(x−y)−2x−2y
2(x−y)
=x−3y
2(y−x)·−1
−1+x+y
2(x−y)−2x−2y
2(x−y)
46. 5(2y+1)
2y−3−3(y−1)
3−2y−3(y−2)
2y−3
=10y+8
2y−3
48. (x+ 1)(2x−1)
(2x−3)(x−3) −(x−3)(x+1)
(3 −x)(3 −2x)+
(2x+ 1)(x+3)
(3 −2x)(x−3)
50. 4t
t(t+ 1)(t−1)
52. 1
x−y−2x
x2−y2+1
x+y
54. 2b
a2−b2−1
a+b+1
a−b
=2b−(a−b)+(a+b)
(a+b)(a−b)
58. 2.5x+15.5=0.5+4x
Chapter 6 Mid-Chapter Review
3. True; see pages 423 and 431 in the text.
6. x−1
=x−1
x−2·x+2
x+2−x+1
x+2·x−2
x−2−6−x
(x−2)(x+2)
=x2+x−2
=3x−6
(x−2)(x+2)
7. t2−16
3
x2−11x+24=0
2w−7=0
2w=7
Chapter 6 Mid-Chapter Review 143
10. x2+2x+3
x2−9=(x+ 1)(x+3)
(x+ 3)(x−3)
11. 6y2+12y−48
3y2−9y+6 =6(y2+2y−8)
3(y2−3y+2)
=2(y+4)
y−1
12. r−s
s−r=−1(−r+s)
s−r=−1(s−r)
s−r
14. x2−100 = (x+ 10)(x−10)
15. a2−a−2
a2−a−6÷a2−2a
2a+a2=a2−a−2
a2−a−6·2a+a2
a2−2a
16. 3y
y2−7y+10−2y
y2−8y+15
=3y2−9y−2y2+4y
(y−2)(y−5)(y−3)
17. x2
x−11 +121
11 −x=x2
x−11 +121
11 −x·−1
−1
=x2
x−11 +−121
x−11
=(x+y)✏(x−y)✏·1
(x−y)✏(x−y)(x+y)✏
=3a−b
a2b·b
b+a+2b
ab2·a
a
20. 5x
x2−4−3
x+4
x+2
=5x
22. 1
x+3−2
x−2=1
x+3·x−2
x−2−2
x−2·x+3
x+3
23. 2
x−2−1
x+3 =2
x−2·x+3
x+3−1
x+3·x−2
x−2
25. 2
x−2+1
x+3 =2
x−2·x+3
x+3+1
x+3·x−2
x−2
26. 2
x−2·1
x+3 =2
(x−2)(x+3)
27. If the numbers have a common factor, their product con-
tains that factor more than the greatest number of times it
for error.
30. Their sum is zero. Another explanation is that
32. The binomial is a factor of the trinomial.
Exercise Set 6.6
2. 6−3
8
4+5
6
=
8−3
8
24
6+5
6
Exercise Set 6.6 145
4. 2+2
3
=
6
3+2
3
6
=2
6.
3
4+7
8
2
=
3
4+7
8
2
·24
24
8. 2−1
a
4+ 1
a
=
2−1
a
4+ 1
a
·a
a
10.
y+1
2y
=
y+1
2y
·2y
=5
3y2
12. 3+2
t
3−2
=
3+2
t
3−2
·t
t
=
2m
m2−9
3m
16.
1
q2−1
1
=
1−q2
q2
1+q
18.
t
4+1
t
=
t
4t
t+1
t
4
20.
1
x2−1
y2
2
=
1
x2−1
y2
2
·x2y2
=(y+x)(y−x)✏
22. x−3+ 2
x
x−4+ 3
x
=
x2
x−3x
x+2
x
x2
x−4
x+3
x
x2−3x+2
x
24.
5
x3−1
x2
2
=
5
x3−1
x2
2
·x3
x3
26.
5
4x3−3
8x
3
2x+3
4x3
=
5
4x3·2
2−3
8x·x2
x2
3
28. a
b−c
d
b
a−d
c
=
a
b·d
d−c
d·b
b
b
a·c
c−d
c·a
a
30.
a
6b3+4
9b2
5
6b−1
9b3
LCM of the denominators is 18b3
32. x−7+ 5
x−1
x−3+ 1
x−1
=
x−7+ 5
x−1
x−3+ 1
x−1
·x−1
x−1
3b−24 >−6b−2
9b>22
b> 22
9
=
2x
x−1
2
x−1
5
Exercise Set 6.7 147
40.
z
1−z
2+2z
−2z
2z
5z−2−3
=
z
2+2z−z
2+2z
−2z
2z−15z+6
5z−2
5z−2
−2z
Exercise Set 6.7
4. 2
3+5
6=1
x,LCM = 6x
6x2
3+5
6=6x·1
x
6. 3
5+2
3=x
9
8. 1
x=1
8−3
5
24t1
8+1
12=24t·1
t
3t+2t=24
5t=24
148 Chapter 6: Rational Expressions and Equations
14. x
5−5
x=0
16. 4
x=5
x−1
2
2x4
x=2x5
x−1
2
18. 5
2y+8
y=1
20. x−7
x+2 =1
4
4(x+2)
x−7
1
22. 8
y−3=6
y+4
(y−3)(y+4)·8
y−3=(y−3)(y+4)·6
y+4
24. x
8−x
12 =1
8
26. x+1
3−x−1
2=1
This checks.
28. x−7
x−9=2
x−9
Check:
x−7
30. x+7
8x−5=2
3
13 =x
This checks.
32. y+11
y+8 =3
y+8
This value does not check. There is no solution.
34. 6
y=5
y−8
Exercise Set 6.8 149
36. t+5
t−2=t−2
t+4
4x+12+2x=x−3
5x=−15
40. 5
y−3−30
y2−9=1
42. 3
x−7=x+10
x−7
44. 5
x−1+x+1= 5x+4
x−1,LCM = x−1
=(x−1) ·5x+4
x(x−5)=0
46. x2
x2−4=x
x+2−2x
2−x
48. 3x−9
x−3=5x−4
2
0=5(x−3)(x−2)
x=3or x =2
y2−4=2y+6−y+2
y2−4=y+8
a denominator zero.
52. 0
Exercise Set 6.8
2. Familiarize. We complete the table shown in the text.
d=r·t
Distance Speed Time
150 Chapter 6: Rational Expressions and Equations
From the rows of the table we have two equations.
330 = (r−14)t,
400 = rt
Then t=330
r−14 and t=400
r,so
4.
Cheetah 10 r+28 t
From the rows of the table we have two equations.
Then r+ 28 = 70.
The gray fox’s speed is 42 mph, and the cheetah’s speed is
70 mph.
6.
Distance Speed Time
Lexus 120 r+30 120
r+30
Then r+30=80.
8.
Distance Speed Time
Slow trip 126 r t
Fast trip 126 r+8 t−1
0=r2+8r−1008
0=(r+ 36)(r−28).
10.
Distance Speed Time
12.
Distance Rate Time
Hobart 20 r t +1
Evan 15 r t
14. Let t= the number of minutes the job would take, working
16. Let t= the number of hours it takes them to do the job,
y x 4
2
18. Let t= the number of days it takes them to do the job,
20. Let t= the number of hours it takes to do the job, working
together.
22. Let t= the number of minutes it would take the two ma-
chines to make one copy of the manual, working together.
24. 800 mi
50 gal =16mi
gal
30. Solve: K
42 =234
14
34. Solve: 60
3000 =S
5000
38. Solve: 3
20 =P
63,240
44. Solve: r
10 =6
8
10
r=8
6,r
6=10
8,6
r=8
10,r
10 =12
16,10
r=16
12,
n=25
4,or 6.25
(One of the following proportions could also be used:
W= 90 whales
50. m=−11 −3
58.
Distance Speed Time
From the rows of the table we have
30 = rt,
Exercise Set 6.9
2. y=kx
4. y=kx
3=k·33
0.8=k·0.2
4=k
8. R=kW
R=0.014W
10. f=kF
12. d=kw
40 = k·3
14. M=kE
38 = k·95
0.4=k
16. y=k
x
18. y=k
x
20. y=k
x
0.3
0.54 = k
22. t=k
r
24. W=k
F