Section 7.6 Solving Rational Equations
49. 11 1
pq f
+=
11 1
fpq fpq
pq f
+=
50.
11 1
pq f
+=
51.
()()()
1
11
1
A
Pr
A
rP r r
=+
+=+
+
PPr A
Pr A P
Pr A P
PP
+=
=−
−
=
Chapter 7 Rational Expressions
53.
12
2
Gm m
Fd
=
54.
12
2
2
12
2
2
1
Gm m
Fd
Fd Gm m
Fd
mGm
=
=
=
56.
xx
zs
sz x x
xx
sz
−
=
=−
−
=
R
58.
E
I
R
r
IR Ir E
Ir E IR
EIR
rI
=+
+=
=−
−
=
Section 7.6 Solving Rational Equations
60.
()
12
12
1212
112 2
121
1
2
1
ff
fff
ff ff f f
ff f f ff
ff f f f
ff
fff
=+
+=
=−
=−
=−
61. Solve
2
2
10 7
15
20
x
x
xx
−=+ −
−−
1.
2
−
Factor the first denominator.
Chapter 7 Rational Expressions
62. Solve
2
2
42 4
14
28
xx
x
xx
+−
=+ −
−−
.
Factor the first denominator.
()()
2
42 4
1
42 4
xx
xx x
+− =+
−+ −
63. Simplify
2
2
10 7
15
20
x
x
xx
−−− −
−−
.
Factor the first denominator.
Section 7.6 Solving Rational Equations
64. Simplify
2
2
42 4
14
28
xx
x
xx
+−
−− −
−−
.
65. Solve
21
2
516
56
1
yy
y
y
−−
+=
+=
Restrictions: 0y≠
LCD =
2
y
Chapter 7 Rational Expressions
66. Solve
21
2
314
34
1
yy
y
y
−−
+=
+=
Restrictions: 0y≠
67. Solve
2
31 10
11 1
yy
y
−=
+− −
.
Factor the denominators.
()
()( )()()
11
310
111 1 1
yyyy
−⋅
−=
+−− + −
Section 7.6 Solving Rational Equations
68. Solve
4125
22 6
4125
22 6
yyy
yy y
−=
−− +
+=
−− +
69.
400 500,000 ;450
x
CC
x
+
==
400 500,000
450 x
+
=
70.
400 500,000 ;405
x
CC
x
+
==
400 500, 000
405 x
x
+
=
LCD = x
Chapter 7 Rational Expressions
71.
2;2
100
x
CC
x
==
−
2
2100
x
=−
72.
2;8
100
x
CC
x
==
−
2
8100
x
=−
73.
; 300, 1000
12
DA
CCD
A
===
+
1000
300 12
A
A
=+
LCD =
12A+
A
74.
; 500, 1000
12
DA
CCD
A
===
+
1000
500 12
A
A
=+
75.
10,000 3 ; 350CxC
x
=+=
10,000
350 3x
x
=+
LCD = x
200
3
2
66 67
3
x
=
=≈
For yearly inventory costs to be $350, the owner
Section 7.6 Solving Rational Equations
76.
10,000 3; 790
CxC
x
=+=
10,000
790 3
x
x
=+
LCD = x
77. – 82. Answers will vary.
83. does not make sense; Explanations will vary. Sample explanation: The sum of two rational expressions is an expression
and thus does not have a solution set.
87. false; Changes to make the statement true will vary. A sample change is:
11 61 1 6 6 .
66 66 6 6
xxx
xxxxxx
+
+=⋅+ = + =
88. true
Chapter 7 Rational Expressions
91.
()()()()()()
222
1726
2115295 25
1726
21 5 21 5 5 5
xxx
xx xx x
xxx
xx xx x x
+−−
=−
−+ +− −
+−−
=−
−− −+ + −
Restrictions:
1,5, 5
xxx≠≠≠−
92.
()
()
()
()
24
24
24
24
11
0
77
17
0
71
17
0
71
xx
xx
xx
xx
xx
xx
++
÷=
++
++
⋅=
++
++
⋅=
++
Restrictions:
7, 1
xx≠− ≠−
Section 7.6 Solving Rational Equations
93. 74
13
xx
b
++=
If x = −6, this equation because
()
76 4
13 6.
b
−++=−
94. 6
24
xx
+=
The solution set is
{
}
8.
95. 50 2x
x=
The solution set is
{
}
5, 5 .−
96. 65xx
+=−
{
}
97.
43
236xxx+−−
Factor by grouping.
()
()
43
43
236
236
xxx
xx x
+−−
=+ +−−
99.
()
[]
[]
54 2 3 54 8 3
54 11
xx
x
−−−=−−−
=− −
15(8 ) 9(8 )
120 15 72 9
24 48
2
xx
xx
x
x
−= +
−=+
−=−
=
The solution set is
{
}
2.
101. In 1 hour you can complete
1
of the job.
Chapter 7 Rational Expressions
7.7 Check Points
1. Let x = the rate of the current.
Then
3
x+=
the canoe’s rate with the current.
and
3
x−=
the canoe’s rate against the current.
Distance
Distance Rate Time Rate
10
With the current 10 3 3
xx
=
++
2. Let x = the number of hours for both people to paint a house together.
Fractional part Time Fractional part
of job completed working of job completed
in 1 hour together in hours
1
First person 88
x
x
x
Working together, the two people can complete the whole job, so
1.
84
xx
+=
Multiply both sides by the LCD, 8.
Section 7.7 Applications Using Rational Equations and Proportions
3. Let x=the property tax on the $420,000 house.
Tax on $250,000 house Tax on $420,000 house
Assessed value ($250, 000) Assessed value ($420, 000)
=
4. Let x=the total number of deer in the refuge.
120 25
150
x
=
5. 312
8
x
=
6. 56
23.5
3.5 2 56
h
h
=
=⋅
7.7 Concept and Vocabulary Check
1. distance traveled; rate of travel
Chapter 7 Rational Expressions
7.7 Exercise Set
1. The times are equal, so 10 15
3xx
=+
To solve this equation, multiply both sides by the LCD,
()
3xx+.
2. The times are equal, so 40 15
20xx
=
+
To solve this equation, multiply both sides by the LCD,
()
20xx+.
3. Let x = the jogger’s rate running uphill.
Then x+ 4 = the jogger’s rate running downhill.
Distance
Distance Rate Time Rate
=
Section 7.7 Applications Using Rational Equations and Proportions
4. Let x = the car’s rate.
Then x−20 = the truck’s rate.
Distance
Distance Rate Time Rate
120
Truck 120 20 20
xx
=
−−
5. Let x = the rate of the current.
Then 15 x+ = the boat’s rate with the current.
and 15 x− = the boat’s rate against the current.
Distance
Distance Rate Time Rate
20
With the current 20 15 15
xx
=
++
Chapter 7 Rational Expressions
6. Let x = the rate of the current. Then 18 x+ = the boat’s rate with the current and 18 x− = the boat’s rate against the
current.
Distance
Distance Rate Time Rate
33
With the current 33 18 18
xx
=
++
7. Let x = walking rate.
Then 2x = jogging rate.
Distance
Distance Rate Time Rate
2
Walking 2
2
Jogging 2 2 2
xx
xx
=
8. Let x = running.
Then 9x = driving rate.
Distance
Distance Rate Time Rate
5
Running 5
xx
=
Section 7.7 Applications Using Rational Equations and Proportions
The total time is 3 hours, so 5903
9xx
+=
or 5103.
xx
+=
To solve this equation, multiply both sides by the LCD, x.
9. Let x = the boat’s average rate in still water.
Then 2x+ = the boat’s rate with the current (downstream).
and 2x− = the boat’s rate against the current (upstream).
Distance
Distance Rate Time Rate
6
Downstream 6 2 2
xx
=
++
10. Let x = the canoe’s average rate in still water. Then 2x+ = the canoe’s rate with the current and 2x− = the canoe’s
rate against the current.
Distance
Distance Rate Time Rate
6
Downstream 6 2 2
xx
=
++
Chapter 7 Rational Expressions
11. Let x = the time in minutes, for both people to shovel the driveway together.
Fractional part Time Fractional part
of job completed working of job completed
in 1 minute together in minutes
1
You 20 20
x
x
x
12. Let x = the time in minutes, for both people to wash the car together.
Fractional part Time Fractional part
of job completed working of job completed
in 1 minute together in minutes
1
You 40 40
x
x
x
13. Let x = the time, in hours, for both teams to clean the streets working together.
Fractional part Time Fractional part
of job completed working of job completed
in 1 hour together in hours
1
First team 400 400
x
x
x
Section 7.7 Applications Using Rational Equations and Proportions
Working together, the two teams complete one whole job, so 1.
400 300
xx
+=
Multiply both sides by the LCD, 1200.
14. Let x = the time, in hours, for all three crews to dispense food and water working together.
Fractional part Time Fractional part
of job completed working of job completed
in 1 hour together in hours
1
First team 10 10
x
x
x
Working together, the two teams complete one whole job, so 1.
10 15 20
xxx
++ =
Multiply both sides by the LCD, 60.
15. Let x = the time, in hours, for both pipes to fill in the pool.
Fractional part Time Fractional part
of job completed working of job completed
in 1 hour together in hours
1
First pipe 44
x
x
x
Chapter 7 Rational Expressions
16. Let x = the time, in hours, for both pipes to fill in the pool.
Fractional part Time Fractional part
of job completed working of job completed
in 1 hour together in hours
1
First pipe 33
x
x
x
17. Let x = the tax on a property with an assessed value of $162,500.
Tax on $62,000 house Tax on $162,500 house
Assessed value ($62,000) Assessed value ($162, 500)
$720 $
x
=
=