Math 080 Final Exam Review – 2008
Note: If you have difficulty with any of these problems, get help, then go
back to the appropriate sections in the text and work more problems!
For problems 1 through 20 , solve for x, writing solutions of inequalities in
interval notation:
2
3
4
1


1.
2.
2x  5  4  1 x  4   7
2
x 1 x 1 x 1
5
2
10
3. 1 – 3x < 7
2x  3  6  1
4.
5. a  2 xt
6.
7. 2 x 2  x  15
8. ln x  3
9. x 2  2x  2  0
10.  6  2x 1  1
11. 8 x  1
12.
3
3
5x 7
16
13. log8 (2x)  log8 (x  3)  1
15. 3x 4  x 2  2  0
18.
5
3

1
x x2
14.
4x  1  5
16. 2 x 1  3
19.
3  x 1  x
17.  3x 2  4 x  9  5
x
2x  1
  4 or
2
2
3
20.  2e x  24
For problems 21 through 23, solve each system:
21. 5x – 2y = 4
22. x + y – z = 4
2x + 3y = 13
3x – 5y + 4z = 3
6x – 7y – 2z = 2
23. x + y + z = 57
3x + y
=3
x
–z=6
For problems 24 through 47, perform the indicated operations and/or
simplify as much as possible:
24. (8x  1)
2
2
26. (x  1)( x  x  1)
25. (8x 3  9x 2  6x)  (4 x 2  11x  6)
27.
1 a
1
x2  1
x4
29. 2

x  2x  1 x  1
31.
x 2 x 5

x 3 3x
b
a2
 4 x 2 (3x 6 )
b2
30.
32.
(3x 2 )2 (2x5 )
28.
6
2
m m
x 2  x  12
x 3  9x 2  20x


2
2
m  2m
x3  x2y
x2  y2
33.
2
3

a 1 a  2
 6 x 2  13x  5 


 5x 2  x



35.
 2x  5 


 5x  1 
34. 2 3
38. 8  20  2  500
41.
 x 1  y 1 2
3
1
44. (16) 4 (27) 3
39.
36.
m 10
m
x 2  10x  25
37. 4 32  3 50
40. (8)2 (2a) 3
3
 25x5  1
2
42. 
 16x 3 


45.
14 2
43. (4 x 3  8x 2  9x  1)  (2x  3)
46.
3
80a7b11
47.
4
32a8b3
For problems 48 through 51, simplify by rationalizing the denominator:
3
6
2x
5 7
2a
48.
49.
50.
51.
3 2
7 2
6x
5 7
3b
For problems 52 through 54, find the exact value of each logarithm:
52. log3 81
53. log100
54. log4 1
8 
55. Find, algebraically, the vertex of the parabola f ( x)  2x 2  4 x  5 . Find the
maximum or minimum value of the function and indicate whether the value is the
maximum or whether it is the minimum.
56. Find, algebraically, the x-intercept(s) of the graph of f(x) =  2x 2  3x  1 .
Give exact, simplified values and then round your answers to two decimal places.
57. Solve by completing the square: x 2  9x  5  0 .
58. Find the domain (in interval notation) of
f (x) 
7x
.
4  x2
59. Find the domain (in interval notation) of g(x)  4 x  3 .
For problems 60 through 65, factor completely:
60. x 4  3x 3  8x  24
63. 2x 2  7x  4
61. 25x 2  40xy  16y 2
64. 64 x 2  121y 4
62. 1000x 3  y 3
65. 54c 3  16d3
***For problems 66 through 71, graph on a rectangular coordinate system
(use graph paper). Additionally, for problems 67, 68, 69 and 70, give the
domain and range of each function by reading it from your graph.
66. 3x – 4y = 12
67. f (x)   (x  2)2  9
68. g(x) = 3
69. p( x)  2x  3
70. F(x)  x 2  3x  2
71. x = – 2
72. Find the slope and y-intercept of 3x – 2y = 2.
73. Find the slope of the line passing through the points (– 6,4) and (8,3).
74. Write the equation of the line passing through the point (– 2,5) and
a) parallel to 3x + 2y = 5
b) perpendicular to 2x – 4y = 12.
75. If f(x) = x 2  7 and g(x) = 2x + 3, find (f  g)( x) . Simplify.
76. If f (x)  1 x  3 , find a formula for f 1 ( x ) .
2
x  x2
77. Find f(5) if f ( x ) 
. Simplify.
x7
78. Find f(x + h) if f (x)  3x  1 . Simplify.
79. Use properties of logarithms to write as a single logarithm:
5 log a  3 log b  1 log c .
2
80. Use properties of logarithms to write as a sum/difference of simpler logarithms:
 3x 2 

log2 
 y 


For problems 81 and 82, perform the indicated operations with the complex
numbers:
1  3i
81. (7 – 4i)(5 + 4i)
82.
4i
For problems 83 through 100, use algebraic methods consistent with
what your instructor expects.
83. An airplane can travel 300 miles against the wind in the same time it travels
400 miles with the wind. If the speed of the wind is 25 mph, what is the speed
of the plane in still air?
84. Using straight line depreciation, the value V of a particular piece of office
equipment x years after purchase is given by the linear function
V(x) = 5,400 – 675x for 0  x  8 .
a) Identify the independent and dependent variables and explain what
each represents.
b) What is the domain of this function?
c) What is the value of this office equipment 3 years after it is purchased?
d) After how long will the value of this equipment be $0?
85. An object falls from a window 400 feet high. Its distance above the ground is
given by the function d(t)  400  16t2 , where t is in seconds and d(t) is in feet.
a) Find the height of the object after 2 seconds.
b) How long does it take the object to hit the ground?
86. Find the dimensions of a rectangular floor if its width is 1 foot less than half its
length and if the floor can be covered by 112 square feet of carpeting.
87. Suppose that a 750 square foot apartment rents for $1550 and that a 1050
square foot apartment rents for $1925. Suppose that the relation between
the size (number of square feet) of an apartment and rent is linear.
a) Find a linear function that expresses the cost C of renting an apartment
as a function of its size s.
b) Interpret the slope.
c) Predict the rent of a 900 square foot apartment.
d) If the rent of an apartment is $1800, how big (number of square feet)
would we expect it to be?
88. A student drives 20 miles at a constant speed, but she will be late for her
Intermediate Algebra final exam unless she drives 20 mph faster for the last
30 miles. If her total travel time is 1 hour, what is her original speed?
89. Using data from 1960 to 2000, the function N(x)  271.40x  836.83
represents the approximate number of registered climbers at Mt. Rainier
x years after 1960. (Source: National Parks Service, U.S. Dept of the
Interior)
a) Identify the independent and dependent variables and explain what
each represents.
b) Evaluate N(43) and explain what it represents.
c) There were 9714 registered climber on Mr. Rainier in 2003. Compare this
value to your result in part (b) and comment on any differences.
90. A man rows 20 miles downstream and return in 13 hours and 20 minutes.
If he can row 4 mph in still water, what is the rate of the current?
91. A businessman received a total of $1240 in interest for the year on two investments. If one of the investments paid 7% annual interest and the other paid
12% annual interest, how much was invested at each rate of the total amount of
money invested was $12,800?
92. If each works alone, it takes one clerk twice as long as another clerk to stuff a
batch of envelopes. Together, the two clerks can do the job in 3 hours. How
long does it take each clerk, alone, to stuff the envelopes?
93. A company offers two types of health plans to its employees. Plan A pays 90%
of an employee’s medical bills after a $400 deductible, while Plan B pays 80%
of an employee’s medical bills after a $250 deductible. For what amounts of
medical bills is Plan A better for an employee than Plan B?
94. A new car has a sticker price of $32,590. Suppose that the dealer markup on
this car is 15%. What was the dealer’s cost, to the nearest dollar?
95. The outside dimensions of a picture frame are 40 inches by 32 inches.
The area of the picture within the frame is 1008 square inches. Find the
width of the frame.
96. An investor split $60,000 among three banks. He received 5%, 6%, and 7%
in interest on the three deposits. In the account earning 7% interest, he deposited
twice as much as in the account earning 5% interest. If his total earnings were
$3760, how much did he deposit in each bank?
97. It takes 2 hours for a boat to travel 28 miles downstream. The same boat can
travel 18 miles upstream in 3 hours. Find the speed of the current and the speed
of the boat in still water.
98. A small company found that when their product is sold for a price of x dollars,
the weekly revenue in dollars as a function of the price x is R(x)  0.25x 2  170x .
For what selling price will the weekly revenue be maximized, and what is the
maximum weekly revenue?
99. Based on data from the Kelly Blue Book, the value V of a Dodge Stratus that is t
years old can be modeled by the function V(t)  19,282(0.84) t . According to the
model, when will the car be worth $12,000? Round answer to the nearest
hundredth.
100. A wood craftsman makes children’s rocking horses. He sells each rocking
horse for $95. His monthly fixed costs of operating his business are $4500, and
it costs $35 in materials for each rocking horse.
a) Find the revenue function R treating the number of rocking horses x as
the independent variable.
b) Find the cost function C treating the number of rocking horses x as the
independent variable.
c) Find the break-even number of rocking horses that must be manufactured
and sold. What is the revenue and cost of this number of rocking horses?
Solutions:
1. – 63
2. no solution
8. e 3
7. 5 ,  3
2
19. (,8] 
72 , 
32.
x ( x  3)
( x  5)( x  y )
2
42.
5x 4
4
4
56.
53. 2
21. (2,3)
22. (3,2,1)
33.
5a  1
(a  1)(a  2)
1
39.
8
2 x 3
2
49.
35.
8
8
3
36. 16
x 2  2xy  y 2
3
46. 2a2b3 10ab 2
x 5
45.
50.
3x  1
x
x2y2
41.
a3
44. 24
6x
3
2x  3
x3
31.
34. 1
40. 
m7
48. 2 7  4
54.  3
b
ba
27.
4m  14
m(m  1)(m  2)
30.
43. 2 x 2  x  3 
47. 2a2 2b3
52. 4
26. x 3  1
38.  4i 5
37.
( 2.485)
18. 3  19
24. 64 x 2  16x  1
x 1
x4
29.
12. 2
3
 ,2  [ 23 , )
17.
( .585)
20. ln12
25. 8x 3  5x 2  17x  6
3x
2
11.  4
3
23. (– 60,183,– 66)
28. 
6. (– 2,12)
2t
15.  1,  i 6
2
log 3
 1 or log2 3  1
log 2
5. 3a
4. 4, – 1
,1]
10. ( 19
2
9. 1  i
14. (, 3 ]  1,  
13. 1
16.
3. (2, )
18ab
3b
51.
16  5 7
9
55. vertex = (1,7), maximum value = 7
3  17
3  17
and
; approximately –.28 and 1.78
4
4
57.  9  101
2
2
58. (,2)  (2,2)  (2, )
60. (x  3)( x  2)( x 2  2x  4)
63. (2x + 1)(x – 4)
61. (5x  4 y)2
59. [ 3 , )
4
62. (10x  y)(100x 2  10xy  y 2 )
64. (8x  11y 2 )(8x  11y 2 )
65. 2(3c  2d)(9c2  6cd  4d2 )
For problems 66 through 71, see instructor for graphs.
72. slope = 3 , y-intercept = – 1 or (0,– 1)
2
74a) y   3 x  2
b) y = – 2x + 1
76. f 1 ( x)  2x  6
77. 10
2
80. log2 3  2 log2 x  1 log2 y
2
73.  1
14
75. (f  g)( x)  4 x 2  12x  2
78. 3x + 3h + 1
81. 51 + 8i
 a5 

79. log
 b3 c 


82. 1  13 i
17
17
83. speed is 175 mph
84. a) independent variable is x, which represents the time in number of years;
dependent variable is V, which represents the value of the office equipment
b) [0,8]
c) $3375
d) after 8 years
85. a) 336 ft b) 5 seconds
86. length is 16 ft, width is 7 ft
87. a) C = 1.25s + 612.5
b) slope = 1.25, and this means that the cost of
renting an apartment is increasing by $1.25 for each square foot increase in its size
c) $1737.50
d) 1930 square feet
88. original speed = 40 mph
89. a) independent variable is x, which represents the number of years since 1960;
dependent variable is N, which represents the number of registered climbers at Mt.
Rainier b) approximately 12,507; this represents the number of registered climbers in
the year 2003
90. rate of current = 2 mph
91. $5920 at 7% and $6880 at 12%
92. 4.5 hours and 9 hours
93. medical bills must exceed $1600
94. dealer’s cost was $28,33 95. width is 2 inches
96. $16,000 at 5%, $12,000 at 6%, $32,000 at 7%
97. speed of current = 4 mph and speed of boat in still water = 10 mph
98. selling price to maximize revenue is $340; maximum revenue is $28,900
99. in about 2.72 years
100. a) R(x) = 95x b) C(x) = 35x + 4500 c) 75 rocking horses; $7125