Chapter 7 Review Exercises
47.
2
12
3
16
1
x
x
+
−
LCD =
2
x
48.
31 1
6xx
−=
The restriction is
0.x≠
The LCD is 6x.
31 1
6
31 1
66
6
{
}
xx
xx
xx
−=
−=
311
44
44
34
1
xx
xx
x
x
=+
=+
−=
The solution set is {−1}.
()()
610
xx
+−=
6 0 or 1 0
6 1
xx
xx
+= −=
=− =
The solution set is
{
}
6,1 .−
51.
5
455
x
xx
−=
++
The restriction is
5.x≠−
The LCD is
5x+
.
Chapter 7 Rational Expressions
52.
2
248
33 9
xxx
=+
−+ −
To find any restrictions and the LCD, all denominators should be written in factored form.
()()
24 8
33 33xx xx
=+
−+ +−
53.
22
36
x
x=+
Restriction:
0x≠
54.
13 1
3
11yy
−=
−−
Restriction: 1y≠
LCD = 1y−
Chapter 7 Review Exercises
55.
()()
2
11 1
31 23
11 1
31 31
x
xxxx
x
xx xx
+
−=
+− +−
+
−=
+− +−
56.
()
250 3 5
25
t
Pt
+
=+
()
250 3 5
125 25
t
t
+
=+
57.
1
C
Sr
=−
140
200 1
r
=−
Chapter 7 Rational Expressions
58.
R
RC
Pn
−
=
59.
11 2 2
12
11 2 2
12 12
12
PV P V
TT
PV P V
TT TT
TT
=
=
60.
A
P
TPr
−
=
()
A
P
Pr T Pr Pr
−
=
61.
12
111
R
RR
=+
12 12
12
111
RR R RR R
RRR
=+
Chapter 7 Review Exercises
62.
nE
I
R
nr
=+
()()()
nE
RnrI Rnr
R
nr
+=+
+
63. Let x = the rate of the current. Then
20 x+=
the rate of the boat with the current and
20 x−=
the rate of the boat
against the current.
Distance
Distance Rate Time Rate
=
The times are equal, so
()()
72 48
20 20
72 20 48 20
xx
xx
=
+−
−= +
64. Let x = the rate of the slower car. Then
10x+=
the rate of the faster car.
Distance
Distance Rate Time Rate
60
Slow car 60
xx
=
Chapter 7 Rational Expressions
65. Let x = the time, in hours, for both people to paint the fence together.
Fractional part Time Fractional part
of job completed working of job completed
in 1 hour together in hours
1
Painter 66
x
x
x
66. Let x = number of teachers needed for 5400 students.
3
50 5400
50 3 5400
x
x
=
=⋅
67. Let x = number of trout in the lake.
Original Number Tagged Deer Number Tagged Deer in Sample
Total Number of Deer Total Number Deer in Sample
112 32
=
=
68.
810
4
x
=
Chapter 7 Review Exercises
69. Write a proportion relating the corresponding sides
of the large and small triangle. Notice that the
length of the base of the larger triangle is
9 ft 6 ft 15 ft.+=
15
x
=
70. Since the profit varies directly as the number of
products sold, we have
.pkn=
Use the given
values to find k.
.
pkn
=
$4935.
71. Since distance varies directly as the square of the
time, we have
2
.dkt=
Use the given values to find k.
2
dkt
=
72. Since the pitch of a musical tone varies inversely as
its wavelength, we have
.
k
pw
=
Use the given values to find k.
1056
k
pw
k
=
=
The equation becomes
1056 .pw
=
When
2.4w=
,
1056 440.
2.4
p==
()
2
28
8
28 64
64 28 64 64
k
k
k
=
=
=
Chapter 7 Rational Expressions
74. Since time varies directly as the number of
computers and inversely as the number of workers,
we have
.
kn
tw
=
Use the given values to find k.
The equation becomes
2.
n
tw
=
75. Since the volume varies jointly as height and the
area of the base, we have
.vkha=
Use the given values to find k.
Chapter 7 Test
1.
2
7
536
x
xx
+
+−
Set the denominator equal to 0 and solve for x.
2.
()( )
()( )
2
2
13
23 3
12 2
32
xx
xx x
xx x
xx
−+
+− +
==
−− −
−+
3.
()
2
45
420 4
xx
xx x
−
−==
22
49
xxx
−−
()()
2
34
xx x
−−
6.
2
2
28 54
39
xxx
xx
+++
÷
−−
2
()()
()()
()()
33 11
5
31
yy yy
yy
−− +−
=−−
Chapter 7 Test
8.
2
2565
33
yy
yy
+−
+
++
9.
22
22
23 45
712 712
yy yy
yy yy
−+ −−
−
++ ++
()()
22
2
23 45
712
yy yy
yy
−+− −−
=
++
10.
5
33
x
xx
+
+−
LCD =
()()
33xx+−
5
x
+
11.
22
26
43 2xx xx
+
−+ +−
()( )( )
()
()( )( )
()()
()( )( )
132
63
132
2263
132
xx x
x
xx x
xx
xx x
=−− +
−
+−− +
++ −
=−− +
33
xx
+
−−
()
()
()
()
15
4
313
45
33
x
xx
x
xx
−+
=+⋅
−− −
−−
=+
−−
Chapter 7 Rational Expressions
14.
2
23 2
3
712
x
x
xx
+−−
−+
()()
()
2712 3 4
31 3
xx x x
xx
−+=− −
−= −
LCD =
()()
34xx−−
15.
2
84
4
16
y
y
y
−−
−
()()
()
216 4 4
41 4
yyy
yy
−=+ −
−= −
LCD =
()()
44yy+−
2
84
4
16
y
y
y
−−
−
()()
84
44 4
y
yy y
=−
+− −
16.
()
22
33
xy xxy
xy x y
−−
÷
++
()
2
2
33
xy xy
xy xxy
−+
=⋅
+−
18.
11
1
xy
x
−
LCD =
xy
11
11
11
11
xy
xy
xy
xy
xx
xy xy yx
−
−
=⋅
⋅− ⋅ −
Chapter 7 Test
20.
()
34
1
5210
34
1
525
y
yy
y
yy
−
−=
++
−
−=
++
21.
()()
2
23
1
11
23
1
111
xx
xxx
=+
−−
=+
−+−
Restrictions:
1, 1xx≠≠−
LCD =
()()
11xx+−
22.
as
Ras
=+
()()
as
asR asas
+=+
+
Chapter 7 Rational Expressions
23. Let x = the rate of the current.
Then
30 x+
= the rate of the boat with the current and
30 x−
= the rate of the boat against the current.
Distance
Distance Rate Time Rate
16
Downstream 16 30 30
xx
=
++
24. Let x = the time (in minutes) for both pipes to fill the hot tub.
1
20 30
xx
+=
LCD = 60
25. Let x = number of tule elk in the park.
200 5
150
5 30,000
6000
x
x
x
=
=
=
There are 6000 tule elk in the park.
Cumulative Review
27. Let C = the current (in amperes).
Then R = the resistance (in ohms).
Step 1
k
C
R
=
Step 2 To find k, substitute 42 for C and 5 for R.
42 5
R
k
=
Cumulative Review Exercises (Chapters 1-7)
1.
() ()
23581
26588
xxx
xxx
−+ = −
−+ = −
2.
()( )
3242612
6121224
18 12 24
xx
xx
x
−−> −
−+ > −
−+>−
3.
2
2
318
3180
xx
xx
+=
+−=
Chapter 7 Rational Expressions
4.
()()
2
2
212
22
4
422
x
xx
x
xxx
+=
−+
−
−= + −
{
}
5.
23
29
yx
xy
=−
+=
To solve this system by the substitution method, substitute
23x−
for y in the second equation.
()
29
22 3 9
xy
xx
+=
+−=
6.
32 2
4518
xy
xy
+=−
−+ =
To solve this system by the addition method, multiply the first equation by 4 and the second equation by 3.
Then add the equations.
12 8 8
xy
+=−
Mid-Textbook Check Point
7.
32 6xy−=
x-intercept: 2
y-intercept: −3
checkpoint: (4,3)
8. Graph 23yx=− + by using its slope, −2 =
2
1
−
, and
its y-intercept, 3.
9. 3y=−
The graph is a horizontal line with y-intercept −3.
12.
12
1
4
x
x
−
−
1
4
12
41
xx
x
x
x
⋅−⋅
−
=−
13.
()()
2
413341 3xx xx−+=− −
15.
()
()()
22
3753 25
35 5
xx
xx
−= −
=+ −
16.
()()
22
432576xx xx−+− −−
Chapter 7 Rational Expressions
18.
62 1
23
xx
xx
++
+
−+
LCD =
()()
23xx−+
62 1
xx
++
+
19. Let x = the amount invested at 5%.
Then
4000 x−
= the amount invested at 9%.
()
0.05 0.09 4000 311
xx
+−=
20. Let x = the length of the shorter piece.
Then
3x
= the length of the larger piece.
368
468
17
xx
x
x
+=
=
=
Mid-Textbook Check Point
1.
() ( )
24 2 532 1xx−+=− +
24 856 3
46 62
xx
xx
−−=−−
−−=−+
2.
3
25
−=
Multiply both sides by the least common
denominator of the fractions, 10:
xx
3955
395 555
29 5
299 59
214
xx
xxxx
x
x
x
+≥ −
+− ≥ −−
−+≥−
−+−≥−−
−≥−
4.
23 6
25
xy
xy
+=
+=
Multiply the second equation by
2−
and add the
result to the first equation.
Mid-Textbook Check Point
Back-substitute 4 for y into either of the original
equations to find x. We choose to use the second
equation:
5.
32 1
10 2
xy
yx
−=
=−
Substitute the expression
10 2x−
for y into the first
equation and solve for x.
()
32102 1
xx
−−=
6.
()
34
1
5210
34
1
525
x
xx
x
xx
−
−=
++
−
−=
++
Multiply both sides of the equation by the LCD,
()
25x+
.
7.
65xx
+=−
Multiply both sides of the equation by the LCD, x.
30 or 20
32
xx
xx
+= +=
=− =−
Both proposed solutions check, so the solution set is
{
}
3, 2−−
.
11.
()()
() () () ()
2
2
7435
73 75 43 45
21 35 12 20
21 23 20
xx
xx x x
xxx
xx
+−
=−+−
=−+−
=−−
22 2
Chapter 7 Rational Expressions
14.
()
22
2
6836
xx xx
x
++
÷+
2
22
68 1
36
xx
xxx
++
=⋅
+
15.
()()()()
22
23 54
31 41
xx
xx xx
xx
xx xx
−
+− −+
=−
+− −−
The LCD is
()()()
314xxx+−−
16.
1
5
1
5
x
x
−
−
Multiply the numerator and denominator by the
LCD, 5x.
17.
()()
222
449(2)7
2727
xx
xx
−= −
=+ −
18.
()()
()
()
()()()
32 2
2
33 313
31
311
xxx xx x
xx
xxx
+−−= +−+
=+ −
=+ + −
()
()()
2
422
xx x x
=++−
21.
()
()
32 2
2
10 25 10 25
5
xx xxxx
xx
−+= −+
=−
point
()
0, 1−
. We move up 1 unit and to the right 3
units to the point
()
3, 0
. Find additional points as
needed and connect with a line.
y
5
Mid-Textbook Check Point
24. 32 6xy+=−
Let
0x=
:
()
30 2 6
y
+=−
The x-intercept is
2−
, so the line passes through the
point
()
2, 0−
. Find additional points to check and
connect them with a straight line.
y
5
25. 2y=−
The graph is a horizontal line that passes through
the point
()
0, 2−
.
y
5
26.
()
21
21
33 6 2
213
yy
mxx
−−− −
== ==−
−−−
27.
21
21
62 4 2
31 2
yy
mxx
−−
====
−−
The point-slope equation is
28. Let x = the number.
57208
x
−=
The price of the digital camera before the reduction
is $320.
30. Let x = the width of the field.
3x = the length of the field.
()
2 2 3 400
xx
+=
the field is 150 yards.
31. Let x = the amount invested at 7%.
y = the amount invested at 9%.
Principal Rate Interest
7% 0.07 0.07
9% 0.09 0.09
xx
yy
Multiply the first equation by
0.07−
and add the
result to the second equation.
0.07 0.07 1400
0.07 0.09 1550
xy
xy
−− =−
+=
0.02 150
7500
y
y
=
=
Chapter 7 Rational Expressions
32. Let x = the number of liters of 40% acid solution.
y = the number of liters of 70% acid solution.
No. of Percent Amount
liters of acid of acid
40% Sol 40% 0.4 0.4
xx
=
12
0.4 0.7 6
xy
xy
+=
+=
Multiply the first equation by
0.4−
and add the
result to the second equation.
The chemist should mix 8 liters of 40% acid
solution with 4 liters of 70% acid solution.
33. Let x = the height.
115 120
2
x
⋅⋅=
34. Let x = the measure of the 2
nd
angle.
10x+
= the measure of the 1
st
angle.
420x+
= the measure of the 3
rd
angle.
35. Let x = the price of a TV.
y = the price of a stereo.
3 4 2530
4 3 2510
xy
xy
+=
+=
72590
370
y
y
=
=
Back-substitute 370 for y into one of the original
36. Let x = the width of the rectangle.
6x+
= the length of the rectangle.
()
2
2
655
655
6550
xx
xx
xx
+=
+=
+−=
length is 11 meters.