MTH 104: Intermediate Algebra Course Review
Evaluate the function.
1.
If R  x    x 2  4 x  8 , find R(2)
3.
If g  x   2 x3  5 x , find g (2)
2.
If f  x   x2  3x  7 , find f  2 
Determine whether each relation is a function, explain why or why not, and state its domain and range.
4.
6.
 1,3 ,  2, 2 , 3, 2 ,  4,5 , 5,3 
President
George Washington
Martin Van Buren
Grover Cleveland
Franklin D. Roosevelt
George H. W. Bush
5.
Inauguration year
1789
1841
1885
1893
1933
1989
7.
 1,3 ,  2, 2 ,  2,1 , 3, 4 ,  4,5 
Pepper variety
Bell
Jalapeno
Chipotle
Cayenne
Tabasco
Habanero
Carolina Reaper
Scoville rating
0
1000
3500
30,000
100,000
2,200,000
Simplify. Assume all variables are positive.
8.
23 x
3 x 3 y 1
9.
2
12 15 3
 27x
y

2

10. 16 y
3
4
3
12 4
12. a a


 8 x9 y12 z15  3
11. 

 125 
1
4
1
2 1 8
13.
75a15b12
15.
98
(a b )
14.
3
 250 x 5 y 7
Factor completely.
16. 15xz 2  6 x  5 yz 2  2 y
17. 5 x 2  14 x  8
18. 100 y 2  49
19. 36a 2  42ab  12b 2
20. 125a3  8
21. 64m 4  27mn3
WH, JM, JD, BR, DP
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MTH 104 Course Review
Perform the indicated operation and simplify. Assume all variables are positive real numbers.
22.
3
24
 2
x  5 x  2 x  15
23.
4
3x  4

x 2 x 5
24.
x2  4x
x2
 2
2
x  6 x  8 x  16
25.
x 2  3x  18 x 2  2 x  24

12  x  x 2 x 2  10 x  24
26. 3 3x3  5 x 12 x
3
28.
40 x  3 5 x
27. 5 32  72
29. (2 + √6)(5 − √3)
Simplify.
x 2  36
30. 2
x  3x  18
32.
8
x 2
9
1
33. y 2
3
1
y
5
3
√4x
2

34. y
1

y
31.
4
y 3
3
y 3
Solve the equations for the specified variable.
35. v  at  vo for a
36. mv  mp  bv for v
37. S  a for r
1 r
Solve the equations.
38.
5x  1  4
40. √1 − x = x + 5
42. 6  4  a
a6
a6
39. 5  4 x  8  11
41. 2 3  1  8
x  25 x  5 x  5
43. x 2  196  0
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MTH 104 Course Review
Solve the quadratic equation by completing the square. Express the solution in set notation.
44. x 2  6 x  16  0
45. y 2  10 y  22  0
Solve the quadratic equation by using the quadratic formula. Express the solution in set notation.
47. 20a  25  4a 2
46. 4 x2  3  8x
48. x2  2 x  2  0
Use the discriminant to determine whether the quadratic equation has one real number solution, two real
number solutions, or two complex number solutions.
49. 9 x 2  30 x  25  0
50. 3t 2  t  2  0
Perform the indicated operation and simplify. Write in a  bi form.
51. 5i (3  2i)
52.  3  2i 2
53. 3  2i
7  6i
54.
4
3  7i
Use a calculator to find the value of each of the following. Round your answer to four decimal places.
55. cos 80
56. tan 14.1
Solve using right triangle ABC with C = 90.
57. c = 3 cm, b = 2 cm;
(i) find the exact value of side a
(ii) find the exact values of sin A, cos A, tan A
(iii) find, to the nearest tenth , A and B .
58. A = 28, b = 5; find values of sides a and c, to the nearest tenth.
59. a = 2, b = 5; find the exact values of the three trigonometric functions of angle A.
60. hypotenuse = 11, length of one leg = 7; find the three trigonometric functions of the smaller angle (round
to 4 decimal places)
61. B = 42, a = 9; find all other angles and sides (round to 1 decimal place)
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MTH 104 Course Review
For each graph, determine whether it is a function, and express its domain and range in interval
notation.
62.
63.
5
4
3
2
1
0
0 1 2 3 4 5
5
4
3
2
1
0
0 1 2 3 4 5
Graph the equations. Also determine the axis of symmetry, vertex, x-intercept(s), and y-intercept and
state the Domain and Range.
2
64. y  x  6 x  5
2
65. y   x  4 x  3
Solve the system of equations algebraically (use either substitution or addition/elimination method).
66.
2x  3y  5
x  2y  3
x  y  z  2
68.
2 x  y  z  1
x  2 y  3z  7
70.
72.
74.
2 x  y  5
y  x  6x  7
2
x2  y  1
4 x  y  1
67.
2x  6 y  5
4 x  12 y  5
3x  2 y  4 z  1
69. 5 x  3 y  5 z  2
6 x  2 y  3z  5
71.
73.
x y 2
y  x2  4x  4
y  x 2  4 x  10
y   x 2  2 x  14
y  x2  2
y  2 x 2  6 x  7
For the following word problems, identify the variable(s) used, set up equation(s) and solve algebraically.
75. A 30-foot ladder, leaning against the side of a building, makes a 50 angle with the ground. How far up the
building does the top of the ladder reach? Express your answer to the nearest tenth of a foot.
76. A 70 foot rope is attached to the top of one of the vertical poles used to hold up a circus tent. The other end
of the rope is anchored to the ground 40 feet from the bottom of the pole. What is the height of the tent and
what angle does the rope make with the tent? Express both answers to the nearest tenth (remember about
the units).
77. Maria invests $7500 at 10.4% simple interest for one year. How much additional money must he invest at
a simple interest rate of 14% so that the total interest earned is 12% of the total investment?
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MTH 104 Course Review
78. Flying with the wind, Rob flew 800 miles between Pittsburgh and Atlanta in 4 hours. The return trip
against the wind took 5 hours. Find the rate of the plane in calm air and the rate of the wind.
79. A member of the City Volunteer Corp. can mow and clean up a large lawn in 9 hours. With two members
of the City Volunteer Corp. working, the same job can be done in 6 hours. How long would it take the
second member of the team, working alone, to do the job?
80. How many pounds of gourmet candy selling for $1.80 per pound should be mixed with 3 pounds of candy
selling for $2.60 a pound to obtain a mixture selling for $2.04 per pound?
81. A fenced rectangular area is 300 square feet. If the width is 5 feet less than the length, find the length and
the width of the fenced area.
82. At a business meeting at Panera Bread, the bill for two cappuccinos and three house lattes was $14.55, At
another table, the bill for one cappuccino and two house lattes was $8.77. How much did each type of
beverage cost?
19.6 s
can be used to determine the time t, in seconds, which an object has been falling if
9.8
it has fallen s meters. Suppose an object has been falling for 4 seconds, how far has the object fallen?
83. The formula t 
84. The distance d to the horizon (the farthest point on the ocean that is visible) for a person whose eyes are at
a height h above sea level is approximately d  2Rh , where R is the radius of the earth, and all three
distances are in miles. If the earth’s radius is 3963 miles, how high are your eyes if you can see 10 miles?
85. Jann can travel 5 miles on her rollerblades in the same time Tran Lee can travel 8 miles on his mountain
bike. If Tran’s speed on his bike is 6.3 miles per hour faster than that of Jann on her rollerblades, determine
Jann’s and Tran’s speeds.
86. When sound travels through air (or any gas), the velocity of the sound wave is dependent on the air (or gas)
temperature. The velocity, v, in meters per second, at air temperature, T, in degrees Celsius, can be found
T
by the formula v  331.3 1 
. Find the speed of sound in air whose temperature is 20 C (equivalent
273
to 68 F ) .
87. The height, h, of a fireworks rocket with an initial velocity of 128 feet per second shot from the top of a
200 foot cliff is a function of time given by the equation h(t )  16t 2  128t  200 , where t is the time in
seconds.
(a) How long does it take for the rocket to reach its maximum height?
(b) Find the maximum height of the rocket.
(c) How long does it take for the rocket to hit the ground? (Round your answer to the nearest tenth.)
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MTH 104 Course Review
88.. Sven plans to build a rectangular pig pen beside her barn, centering it along the broad, eastern side of the
barn since pigs love the afternoon shade. She has 88 feet of fencing available. (Note that the side of the pen
that is against the barn will not require fencing. Assume that the side opposite the barn is the length of the
rectangle.)
(a) Find the dimensions of the rectangle that maximize the enclosed area.
(b) What is the maximum area that can be enclosed by the fence?
89. As of 2008, the best-selling movies based on box-offices sales of all time are Titanic, The Lord of the
Rings, and Pirates of the Caribbean. The total sales of all three movies are $4024 million. The combined
sales of The Lord of the Rings and The Pirates of the Caribbean are $354 million more than sales of
Titanic. Sales of The Lord of the Rings are $69 million more than sales of Pirates of the Caribbean.
Determine the sales for each of the movies.
90. One number is seven times as large as another. The sum of their reciprocals is 8. Determine the numbers.
91. The difference of a positive number and three times its reciprocal is 2. Determine the number.
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MTH 104 Course Review
Answers to MTH 104: Intermediate Algebra Course Review
1.
–4
4.
Is a function; domain { 1, 2, 3, 4, 5 }; range { 2, 3, 5 }
5.
Not a function, because 2 appears twice as first value; domain { 1, 2, 3, 4 }; range { 1, 2, 3, 4, 5 }
6.
Not a function, because Grover Cleveland appears twice; domain { GW, MVB, GC, FDR, GHWB };
range { 1789, 1841, 1885, 1893, 1933, 1989 }
7.
Is a function; domain { Bell, Jalapeno, Chipotle, Cayenne, Tabasco, Habanero, Carolina Reaper };
range { 0, 1000, 3500, 30,000, 100,000, 2,200,000 }
8.
2.
 x4 y
24
9.
9x8 y10
13. 5a7b6 3a
12. a 4 b 8
16.
5z
20.
 5a  2   25a 2  10a  4 
22.
3
x 3
2
3.
10. 8 y 9
 2   3x  y 
17.
 x  25x  4
18. 10 y  7 10 y  7 
19. 6  2a  b  3a  2b 

21. m  4m  3n  16m 2  12mn  9n 2
26. 13x 3x
27. 14 2
28.
30. x  6
x3
31. 8 x  16
x4
32.
34. 2 y  6
35. a 
 x  2  x  5
4y  3
38.
3
a  S or S  a
S
S
42. No solution;
43.
4x 6 y8 z10
15. 7i 2
24.
r
11.
14. 5 xy 2  3 2 x 2 y
2
23. 3x  6 x  12
37.
–6
25
1
1
17
14i,14i
39.
v  vo
t
7
25.  x  6
x
 x  4
3
2
x4
29. 10 − 2√3 + 5√6 − 3√2
5x
3
33. 3  y
5 √2𝑥2
36. v  

2𝑥
y
mp
mp
or v 
mb
bm
41.
40. {-3}
6

44. x  2,8 ; 2,8


45. y  5  3; 5  3,  5  3
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MTH 104 Course Review
46. x  2  7 ;  2  7 , 2  7 


47.
49. one real solution
50. two complex solutions
2


2
51. 10  15i


2
5
 
2
48. 1  i
53. 9  32 i
52. 5  12i
85
54. 6  14 i
29
85
29
57. (i) a  5
55. 0.1736
(ii) sin A  5 , cos A  2 , tan A  5
2
3
3
(iii) A  48.2 , B  41.8
56. 0.2512
5 29
2 29 ,
cos A 
29
29
, tan A  2
58. a  2.7 c  5.7
59.
60. sin A  0.6364 , cos A  0.7714 , tan A  0.8250
61. A  48 , b  8.1, c  12.1
62. Not a function; domain (0, 4]; range [0, 4)
63. Is a function; domain (0, 4]; range [1, 5]
64.
65.
y
sin A 
y
5
5
4
4
3
3
2
2
1
-7 -6 -5 -4 -3 -2 -1
-1
1
x
1
2
3
4
5
6
-7 -6 -5 -4 -3 -2 -1
-1
7
-2
-2
-3
-3
-4
-4
x
1
2
3
4
Axis of Symmetry:
x = -3
Axis of Symmetry:
x=2
Vertex:
(-3, -4)
Vertex:
(2, 1)
x-intercept(s):
(-1, 0) and (-5, 0)
x-intercept(s):
(1, 0) and (3, 0)
y-intercept:
(0, 5)
y-intercept:
(0, -3)
Domain: all real numbers (−∞, ∞)
Range: [−4, ∞)
66. 1, 1
 2, 1 ,  6,7 
5
7
Domain: all real numbers (−∞, ∞)
Range: (−∞, 1]
67. Inconsistent system
 no solution
68.
 1, 3, 4
69. 1,  4,  3
1,1 ,  2,0
72.
 0,1 ,  4,17
73.
71.
6
-5
-5
70.
5
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 4, 10 ,  3,11
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MTH 104 Course Review
74.
3,7 ,  1, 1
75. The ladder reaches 23.0 feet up the side of the building.
76, The angle that the rope makes with the tent is 34.8 degrees and the height of the tent is 57.4 feet.
77. Maria invested $6,000 in additional money.
78. The rate of the plane in calm air is 180 mph and the rate of the wind is 20 mph.
79. The second member would take 18 hours.
80. Seven pounds of gourmet candy should be mixed in.
81. The length is 20 feet and the width is 15 feet.
82. Price of a cappuccino is $2.79 & price of a latte is $2.99.
83. 78.4 meters
84. about 0.0126 miles (66.6 feet)
85. Jann’s speed is 10.5 mph, Tran’s is 16.8 mph
86. 343.22 meters per second
87. (a) 4 seconds; (b) 456 feet; (c) 9.3 seconds
88. (a) 44 feet long, 22 feet wide; (b) 968 square feet
89. Titanic sold $1835 million, Lord of the Rings $1129 million, and Pirates of the Caribbean $1060 million
90.
1
and 1
7
91. 3
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