8/17/15
WITH SOLUTIONS
MATH 022ML
TEST 3
REVIEW SHEET
TO THE STUDENT:
This Review Sheet gives you an outline of the topics covered on Test 3 as well as practice
problems. Answers are at the end of the Review Sheet.
I. EXPRESSIONS
A. Evaluate the expression
For problems 1 – 4, evaluate each expression, if possible. Write answers as integers or
simplified fractions.
x
1. 
when x  4
Solutions
3.
64
x 5
# 1-4
2.
4x
x 4
2
when x  2
4.
3
1000
5. Customers are waiting in late at a department store. They arrive randomly at an average
rate of x per minute. If the clerk can wait on 2 customers per minute, then the average
1
time in minutes spent waiting in line if given by T 
for x  2 .
2 x
Solution #5
a) Complete the table
n
0.5
1
1.5
1.9
x
T
b) What happens to the waiting time as x increases, but remains less than 2?
6.
Find any values of the variable that make the expression undefined.
x2
c)
a
a)
x5
1
d) x3  27
b)
x 2 3x  4
solution #6
n
OLD TOPICS:
For problems 7 – 12, evaluate each expression, if possible. Write answers as integers or
simplified fractions.
1
1
bh
when b  3 and h  10
7.
10. 3
2
2
Solution
2
8.  x  3x when x  2
11. 52  54
#7 - 12
9. 32
3
12.  
5
2
B. SIMPLIFY EXPRESSIONS
For problems 13 – 18,
x2  1
x2  4 x  3
x4
14.
12  3x
8 x
15.
x 8
13.
a) state any values of the variable that make the rational
expression undefined, then
b) simplify each rational expression.
4x
16.
4x  4
Solution
b2
17. 2
Solution
#16 - 18
b 4
#13 - 15
3a  a 2
18.
a 2  2a  3
For problems 19 – 28, simplify each radical expression. Do not give a decimal answer.
Assume all variables are positive.
49
24. 64x 4 y
19.
81
25. 28a 6
20.
80
Solution
Solution
3
26.

27
#24 - 27
#20 - 23
21.  32b 2
3
27. 125
5
y
22.
t2
28.
16
90
23.
10
OLD TOPICS:
For problems 29 – 32, simplify each expression.
29. (4 x  2x  1)  (2 x  x  1)
6 x 2  3x
31.
3x
30. (a5b3 )(a4b8 )
32. 4( x  3)  (2 x  1)
2
2
Solution
#29 - 32
C. OPERATIONS ON EXPRESSIONS
For problems 33 – 44, perform the indicated operation on the rational expressions.
Simplify to lowest terms. Leave answers in factored form.
33.
2x  4 6x  9

3x  6
x2
34.
5 x  15 3  x

x2  9
5
u 2  2u 3u  6
37.

u
2u
35.
x2
x2  5x  6

x 2  3x  2
x6
38.
Solution
#33 - 35
36.
x2  x
x

x 3 x 3
x 2  4 5 x  10

x2  x x2  x  2
Solution
#36 - 38
39.
b 1
5  3b
 2
b  2b  1 b  2b  1
40.
41.
42.
x
1

x  4x  4 x  2
m 2  1 2m

m 1 m 1
43.
x
1

x 9 x 3
x
3

x  3x  2 2 x  2
44.
4n
n

3n  2 n  1
2
Solution
#39 - 41
2
Solution
#42, 43
2
2
Solution
#44
For problems 45 – 53, perform the indicated operations on the radical expressions.
Simplify the expressions by factoring out the largest perfect square factors. Assume
that all variables are positive.
45.
15  2 15
46. 2 a  3 a
47.
48.
Solution
#47, 49 - 51
3 12  5 12
51.
75 x 3
3x
52.
6
6
25
53.
8ab
2ab
x 4 x
49. 5 8  128
Solution
#52, 53
50. 4 3  27
OLD TOPICS:
For problems 54 – 58, multiply and simplify each expression.
54. ( x  4)2
57. 8 x  5 x  x  3
55. (3x  2)(2x  5)
58. ( x 1)(2 x2  3x  4)
Solution
#55 - 58
56. (a  3)(a  3)
For problems 59 – 65, factor the expressions completely, or write “prime”.
59. x 2  x  12
63. 2b2  8b  8
60. 2n2  11n  6
Solution
#59 - 63
61. 6 x  13x  5
2
64. 3t 3  18t 2  48t
65. 12 x  10 x  2 x
4
3
2
Solution
#64 - 66
62. x 2  64
66. Is it true that  x  4   x 2  16 ? Explain why or why not.
2
For problems 67 – 68, simplify the exponent expressions by remove all negative and
zero exponents. Assume all expressions are defined.
2
Solution
a 2 b
 6x2 
67. 1 2
#67 - 68
68. 

a b
 8 xy 
II. SOLVING EQUATIONS
A. SOLVING RATIONAL EQUATIONS
For problems 69 – 77, solve each of the following rational equations and find any values
of the variable that make the expression undefined.
69.
70.
71.
72.
73.
3
2

x x3
1 1 1
 
2x 2 x
x
14

1 x5
2x x
1
 
5 10 10
10
5x
x

x2 x2
74.
Solution
#69, 70, 72
75.
76.
77.
3x
12

x4 x4
x x
  8
5 15
1
4 x
3
x3
x3
1
1
2 2
x
x x
2x 
Solution
#73, 74
Solution
#76, 77
B. SOLVING RADICAL EQUATIONS
For problems 78 – 82, solve and check each of the following radical equations
2x  4  4
81.
z 6  z
79. 3  6x  2  11
82.
4 y  y2
78.
80.
Solution #79
Solution #82
x6  x
C. SOLVING WITH TECHNOLOGY
For problems 83 – 89, use your graphing calculator to solve the equations
graphically or numerically:
83. Solve graphically:
x
 x . For your work,
x 1
Solution #83
sketch a graph of each side of the equation on the same coordinate axes.
Circle and clearly label the solution(s) separate from the graph.
84. Solve numerically: 
2
 1  x . For your work,
x
provide a copy of the table use to solve, with the solution row(s)
Solution #84
clearly identified and include a row above and below to solution separate from the
table.
85. Solve graphically: 3x  1  4 . For your work, sketch a graph of each side of the
equation on the same coordinate axes. Circle and clearly label the solution(s) separate
from the graph. Use algebra to solve the equation. Show work.
86. Solve graphically, if possible:
x  2  5 . For your work,
Solution #86
sketch a graph of each side of the equation on the same coordinate axes.
Circle and clearly label the solution(s) separate from the graph.
87. Solve graphically: 3  2x  x . For your work, sketch a graph of each side of the
equation on the same coordinate axes. Circle and clearly label the solution(s) separate
from the graph. Use algebra to solve the equation. Show work.
88. Solve numerically: x 2  2 x  63 . For your work,
provide a copy of the table used to solve,
Solution #88
with the solution row(s) clearly identified and
include a row above and below to solution.
89. Solve graphically: 2 x2  3  5x , by converting to standard from and graphing the
resulting polynomial. For your work, sketch a graph of the equation, then circle and
clearly label the solution(s). Use algebra to solve the equation. Show work.
OLD TOPICS
For problems 90 – 96, solve the following equations or inequality.
90. x 2  25  10 x
91. 2 x3  2 x2  4 x
92. 2 x  7 x  15  0
2
93. 3x  75 x  0
4
2
3
Solution #93 - 95
2  x  x  6
5
95. 7  2  x  1  5  x  1
94.
Solution
#90 - 92
96. 5(2  x)  6  3x .
Also graph the solution set
on a number line and express
in interval notation.
Solution
#96
For problems 97 – 100, solve the systems of equations by either the elimination or
substitutions method.
5 x  5 y  5
97. 
 y  3x  17
2 x  3 y  2
99. 
3x  2 y  16
Solution #99
 x  7 y  12
98. 
3x  5 y  10
3 y  9 x  12
100. 
 y  3x  4
Solution #100
III. FORMULAS
For problems 101 – 107, solve for the indicated variable. Assume there are no zero
denominators.
101.
1 1 1
  for d
12 15 d
102.
V   r 2h for h
103.
P  6a  2b for b
104.
1 1 1
for c
 
a b c
Solution
#101, 103,
104
a T2
for T2
T1
105.
R
106.
1
r
for R.

T Rr
107.
T
D
4V
Solution
#105 - 107
for D
108. Use the distance formula d  ( x2  x1 )2  ( y2  y1 )2 to find the distance between
(2, 4) and (5,10)
IV. APPLICATION PROBLEMS
A. RATIONAL APPLICATION:
For problems 109 – 114, set up and solve a rational equation to find the indicated value.
Round to the nearest tenth.
109. It will take Yansin 8 hours to paint a house alone.
Solution #109
It will take Jared 10 hours to paint the same house alone.
How many hours will it take them working together?
110. It will take Sarita 4 hours to prepare a party. It will take Jasmine 6 hours to prepare the
same party. How many hours will it take them working together?
111. (Optional – ask your instructor) An airplane can travel
380 miles into the wind in the same time that it can travel
420 miles with the wind. If the wind speed is 10 miles per hour,
find the speed without any wind.
Solution #111
112. (Optional – ask your instructor) A boat can travel 114 miles upstream in the same time
that it can travel 186 miles downstream. If the speed of the current is 6 miles per hour,
find the speed of the boat without current.
B. QUADRATIC APPLICATION
113. A baseball is hit into the air and its height h in feet after t seconds can be calculated by
h  16t 2  96t  3 .
a) What is the height of the baseball when it is first hit?
Solution #113
b) What is the maximum height of the baseball?
OLD TOPICS
114. A solution contains 5% salt. How much pure water should be added to 40 ounces of
the solution to dilute it to a 3% solution?
115. A riverboat takes 8 hours to travel 64 miles downstream
and 16 hours for the return trip. What is the speed of the
current and the speed of the riverboat in still water?
Solution #114, 115
116. Monthly average high temperature in degrees Fahrenheit in Columbus Ohio can be
approximated by the polynomial F  1.466 x2  20.25x  9 , where x=1 corresponds
to January, x = 2 to February, and so on. Use your graphing calculator to make a table
using integer inputs x = 1 to x = 12. What is the average high in May?
117. The elevation of Mt. Everest is 8850 meters. Change this elevation to feet. Write your
answer in scientific notation.
1 meter = 39.37 inches
V. GRAPHING
118. For the quadratic equation y   x 2  4 x  12 ,
a) find the y–intercept
b) find the x–intercept
c) find the axis of symmetry
d) find the vertex
e) Use a graphing calculator to check your results
119.
Solution #118 a) - b)
Solution #118 b) - e)
For the quadratic equation y  x 2  6 x  7 ,
a) Find the y- intercept
b) find the x-intercept
c) find the axis of symmetry
d) find the vertex
e) Use a graphing calculator to check your results
OLD TOPICS
121.
3
and passing through ( 1, –3).
2
Label each axis and three other points. Find the equation of the line.
Solution #120
part 2
Solution #120 part 1
Graph 2 x  3 y  6
122.
Find the equation of a line with a zero slope which passes through (4, –2).
120.
On graph paper, draw a line with a slope of 
Graph this equation.
123.
124.
Find the equation of the line passing through (2, –2) and (–1, –5).
Graph this equation and label the y-intercept on the graph.
Solution
#122
Solution #123
Write the point slope equation of the line passing through (5, -4) with slope m  
2
3
Solution #124
ANSWERS TO PROBLEMS
I.
A.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
B.
Expressions
Evaluating Expressions
4 / 9
Undefined
8
–10
a) Table:
X
0.5
1
1.5
1.9
T
2/3
1
2
10
b) T increases greatly
a) all real numbers except 5
b) all real numbers except 4 and -1 ( x  1, 4 )
c) a  0
d) all real numbers
15
–10
–9
8
56 = 15625
25/9
Simplifying Expression
13. x  3, 1; x  1
14. x  4; 1
x3
3
15. x  8; 8  x  1
x 8
16. x  1; x
x 1
17. b  2, 2; 1
b2
18. a  3, 1; a
a 1
19.
20.
21.
22.
23.
24.
7/9
25.
26.
27.
28.
29.
30.
31.
32.
3
4 5
4b 2
y
2
y
3
8x
2a
2
y
7
–3
5
t/4
6x2  x
a9b11
2x 1
2x 13
MATH 022ML
TEST 3
REVIEW SHEET
C. Operations on Expressions
33. 2(2 x  3)
x2
34. –1
35. x  6
x6
36. x  1
37. 2u
3
2
(
38. x  2)
5x
4
39.
b 1
40. m  1
x  6
41.
2( x  1)( x  2)
42. 2 x  2
( x  2) 2
2x  3
43.
( x  3)( x  3)
7 n 2  2n
44.
(3n  2)(n  1)
45. 3 15
46. 5 a
47. 16 3
48. 3 x
49. 2 2
50. 3
51. 5x
52. 6/5
53. 2
Old Topics:
54. x2  8x  16
55. 6 x2  11x 10
56. a 2  9
57. 5x2  23x
58. 2x3  x2  7 x  4
59. ( x  4)( x  3)
60. (2n 1)(n  6)
61. (2 x 1)(3x  5)
62. ( x  8)( x  8)
63. 2(b  2)2
64. 3t (t  2)(t  8)
65. 2 x2 (2 x 1)(3x 1)
66. It is false. Missing middle term.
67. 1
ab
2
68. 16 y
9 x2
II. Solving Equations
A. Solving Rational Equations
69. x 0, x 3, x = 9
70. x 0, x = 1
71. x  –5, x = 2, x = –7
72. x = 1/3
73. x  2, x = 5
74. x 4, x = –3/2
75. x = –60
76. x 3, no solution
77. x 1, x = 0, x = –3/2
B. Solving Radical Equations
78. x = 6
79. x = 11
80. x -2, x = 3
81. x 4, x = 9
82. x 0, x = 3
C. Solving with Technology
83. Solutions: x=0 and x=2

86. There is no solution. The graphs do not
intersect.








































87. Solution: x=1


















88. Numerically: Solutions: x = –7, 9
x
Y1=x2 –2x – 63
8
–15
9
0
10 17
x
–6
–7
–8
Y1=x2 –2x – 63
15
0
17











x
Y1= –2/x Y2 = 1–x
1
–2
0
2
–1
–1
3
–2/3
2
85. Solution: x= 17 / 3  5.6667



89. Solutions: x= –0.5 and x=2
84. Solutions: x= –1 and x=2
x
Y1= –2/x Y2 = 1–x
–2 1
3
–1 2
2
0
Und
1

















Old Topics
90. x  5
91. x  0, 2, 1
92. x  5, 3 / 2
93. x  0,  5, 5
94. x  3
95. x  2






96. x < 2
(–, 2)
117. 8850m  39.37in  1 ft  29035.4 ft  2.90 104 ft
1m
97. (4, –5)
98. (5,1)
99. (–4, –2)
100. Infinitely many solutions
III. Formulas
101. d=20/3
V
102. h  2
r
P
103. b   3a
2
ab
104. c 
ab
RT
105. T2  1
a
106. R=T r + r
12in
V. Graphing
118. a) (0, 12) c = 12; b) (–2,0) and(6,0) ;
c)
x=2
d) (2, 16)
119. a) (0, –7) c = –7; b) (–7,0) and (1,0);
c)
x = -3
d) (–3, –16)
120. y  3   3 ( x  1)  y   3 x  3
2
2





















121. Graph of
2x  3y  6

y
2
x2
3

IV. Application
A. Rational Applications
109. Let x = # hours it will take them working
together
1 1 
x     1 ; x  4.4 hours to finish
 8 10 
110. Let x = # of hours it will take them working
together
1 1
x     1 ; x= 2.4 hours working together
4 6
111. 380  420 ; x=200 mph
x  10 x  10
112. 114  186 ; x = 25mph
x6 x6
113. a) 3 feet initially (t=0);
b) max height is the y-coord. of vertex147 ft.
114. 26.667 gallons of water is needed to dilute it
115. Current: 2mph; Riverboat: 6mph
116. Complete table for 1 to 12
x
1.466 x2  20.25x  9
1
27.784
2
43.636
3
56.556
4
66.544
5
73.6
…
12
40.896
The average high in May (x=5) is 73.6 degrees


107. D=4VT2
108. 85
2





















122. y = –2
y




x









(4, -2)
y = -2




123. y = x – 4
y


y = x-4

x





(2, –2)



124. y  4   2 ( x  5)
3

(–1, –5)


