Review for College
Mathematics
A SELECTION OF PROBLEMS INTENDED TO
MAINTAIN STUDENTS ESSENTIAL ALGEBRA SKILLS
INSTRUCTOR COPY
Equations and Modeling
1
A moving company charges $46 to move a certain machine 10 miles and
$58 to move the same machine 30 miles.
a)
b)
c)
d)
Find an equation that defines this relationship if it is linear.
What will it cost to move the machine 25 miles?
What is the minimum charge for moving the machine?
What is the rate for each mile the machine is moved?
Solution:
a) C = 0.6 m + 14
b) $55
c) $40
d) $0.60 per mile
MATHEMATICS for COLLEGE READINESS
Equations and Modeling
2
The total cost of producing a certain item consists of paying rent
for the building and paying a fixed amount per unit for material.
The total cost is $250 it 10 units are produced and $330 if 30
units are produced.
a) Find the equation that defines this relationship if it is linear.
b) What will it cost to produce 100 units?
c) How much is paid in rent?
d) What is the cost of the material for each unit?
Solution:
a)
C = 4x + 210, where x = number of units produced.
b) $610
c)
$210
d) $4 per unit
MATHEMATICS for COLLEGE READINESS
Equations and Modeling
3
If x2 + 2x = 3, demonstrate algebraically which of the
following is a possible value for x.
a) – 3
–2
c) –1
d) 0
e) 3
b)
Solution:
a) – 3
MATHEMATICS for COLLEGE READINESS
Equations and Modeling
4
How many solutions are there to the systems of
equations? Solve algebraically.
x  2y  6
y x
 1
3 6
Solution:

Infinite: the lines coincide so the solution is
 x, y  : x  2 y  6
MATHEMATICS for COLLEGE READINESS
Equations and Modeling
5
The sum of the square of a number and the product of
the number and 3 equals 40.
a) Write an equation to model this situation.
b) What are the positive answer(s)?
Solution:
a)
b)
n  3n  40
2
n =5
MATHEMATICS for COLLEGE READINESS
Equations and Modeling
6
Two truckers start on a trip at the same time. One heads
west at 60 mph while the other heads east at 55 mph.
Their CB radios work for a distance up to 500 miles
apart. How long will they be able to communicate with
each other? Model and solve algebraically.
Solution:
4 hr 20 min.
MATHEMATICS for COLLEGE READINESS
Equations and Modeling
7
Solve the system of equations algebraically.
3x  4 y  2
y x 1
 
3 4 6
Solution:
 lines are parallel.
empty set: The
MATHEMATICS for COLLEGE READINESS
Equations and Modeling
8
Solve for w: p = 2 ( l + w )
Give your answer in two forms, one answer with a single
fraction and the other as a sum or difference of two
terms.
Solutions:
p  2l
w
2
p
w  l
2
MATHEMATICS for COLLEGE READINESS
Equations and Modeling
9
1
1
 3
Solve for a:
a
b

Solution:
b
a
1  3b
MATHEMATICS for COLLEGE READINESS
Equations and Modeling
10
Solve for x:
1 1 1
 
x y a

ay
Solution: x 
ya
MATHEMATICS for COLLEGE READINESS
Equations and Modeling
11
The Boosters Club held a spaghetti dinner as a
fundraiser. They sold 300 tickets and collected $2200.
If an adult’s ticket cost $8.50 and a child’s ticket cost
$3.50, how many of each were sold?
Solution:
230 adult tickets
70 children’s tickets
MATHEMATICS for COLLEGE READINESS
Equations and Modeling
12
Admission to a school football game was $2 for students
and $3 for nonstudents. How many of each group
attended if there were 900 people and $1920
collected? Model and solve algebraically.
Solution:
120 nonstudents.
780 students.
MATHEMATICS for COLLEGE READINESS
Equations and Modeling
13
Solve algebraically: x2 – 5 = 3x
Solutions:
3  29
x
2
MATHEMATICS for COLLEGE READINESS
Equations and Modeling
14
Bill needs to make 1000 copies before the meeting that
starts in 15 minutes. He begins using a machine that
makes 30 copies per minute after laminates, a newer
machine which copies at 45 pages per minutes become
free . Using both machines, will Bill be able to make all
the copies before the meeting? Explain algebraically.
Solution: No, it will take almost 17 min to make the copies.
MATHEMATICS for COLLEGE READINESS
Equations and Modeling
15
Find the equation of a line with slope of 0 through
(6, 10).
Solution: y = 10
MATHEMATICS for COLLEGE READINESS
Function Theory
16
Which of the following are functions?
a. x3 y = 12
e.
x
 y  3x  4
2
b. x2 + y2 = 20
f.
y(y + 2) = 4 + x
c. x3 + y = 15
g.
d. x (x+2) = 5 + y
h. y = 5
Solution: c, d, e, h
MATHEMATICS for COLLEGE READINESS
x=4
Function Theory
17
If f(x) = x2 – 5, find f(b) – f( c) .
Solution: b2 – c2
MATHEMATICS for COLLEGE READINESS
Function Theory
18
If f(x) = 2x2 – x + 3, find f(x + h) .
Solution: 2 x2  4 xh  2h2  x  h  3
MATHEMATICS for COLLEGE READINESS
Function Theory
19
If f(x) = x2 – 4, find f(a + 5) .
Solution: a2 +10a+21
MATHEMATICS for COLLEGE READINESS
Function Theory
20
If f(x) = x2 – 3 x + 5, find f(x + h) – f(x) .
Solution: 2xh + h2 – 3h
MATHEMATICS for COLLEGE READINESS
Function Theory
21
If f(x) = 3x4 – 4x3 -2x2 +x - 5, find f(-x).
Solution: 2xh + h2 – 3h
MATHEMATICS for COLLEGE READINESS
Expressions with Exponents
22
Simplify:
x y 
 y 1 


Solution:
0 2
1
1
y
3
MATHEMATICS for COLLEGE READINESS
Expressions with Exponents
23
Simplify. Give the answer with positive exponents only:
3
 2  3
 4  
 x   x
0
Solution:
8
x4
MATHEMATICS for COLLEGE READINESS
Expressions with Exponents
24
Simplify. Give the answer with positive exponents only:
6 x 2 y 3 z 4
.
1 2 2
4x y z
3 y5
.
Solution:
6
2x z
MATHEMATICS for COLLEGE READINESS

Expressions with Exponents
25
Multiply
2
5 4
8r
s
3r

 s 
a. 5r 7 s5

b. 5r10s 4
c.
7 5
24r
s
d.
e.
24r10s4
11r 7 s 5 

7 5
Solution: d. 24r s
MATHEMATICS for COLLEGE READINESS
Expressions with Exponents
26
Multiply
 9r
Solution:
9r
1/2
s   r
9/10 5/4
s
MATHEMATICS for COLLEGE READINESS
2/5
1/4
s

Expressions with Exponents
27
Multiply
 16r
Solution:
1/2
s   r
3/4
16r s
MATHEMATICS for COLLEGE READINESS
5/4
s
0

Expressions with Exponents
28
If the area of a circle if  r 2 , what is the area of a circle if
the radius is 3a?
Solution: A  9 a 2
MATHEMATICS for COLLEGE READINESS
Expressions with Exponents
29
4
If the volume of a sphere is  r 3 , what is the volume of a
sphere with radius 3a? 3
Solution:
V  36 a3
MATHEMATICS for COLLEGE READINESS
Expressions with Exponents
30
If the volume of a cylinder is  r 2 h, find the volume of a
cylinder with radius 3x and a height that is twice the
radius.
Solution: V  54 a3
MATHEMATICS for COLLEGE READINESS
Exponential and Logarithmic Functions
31
If
t e
x  3,
a.
3 + ln t
b.
t3
e
then x =
c.

Solution: e. ln t – 3
MATHEMATICS for COLLEGE READINESS
t
3
d.
ln ( t – 3 )
e.
ln t – 3
Exponential and Logarithmic Functions
32
If 7x = 3, then x =
a. 3
7
b.
7
3
Solution:
c.
log 3 7
d.
log7 3
e.
7 
log10 
3 

c. log 3 7

MATHEMATICS for COLLEGE READINESS
Simplifying Expressions
33
Simplify
3x  2 x  x  3  x  4 
a.
21x  4 x 2
b.
27 x  4 x 2
c.
10 x 2  20 x
d.
10 x 2  10 x
Solution: a
MATHEMATICS for COLLEGE READINESS
Simplifying Expressions
34
Simplify
 2a
2
 b    a  2b 
2
2
a. a 2  3b 2
b. a 2  5b 2
2
2
c. a  4ab  5b
2
2
a

4
ab

3
b
d.
Solution: c
MATHEMATICS for COLLEGE READINESS
Simplifying Expressions
35
Simplify 30k  7  k
   4  5k 
Solution: 5k2 + 250k -16
MATHEMATICS for COLLEGE READINESS
2
Simplifying Expressions
36
Evaluate the following expression using x = -3.1 and y = 4.23
4x-2 + y3
Solution: 76.103
MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
37
Factor completely using integer coefficients.
12m4 n3  48m2 n3
Solution: 12 m2n3 (m2 + 4)
MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
38
Factor completely using integer coefficients.
12m2  8m  9m  6 
Solution: (4m – 3) (3m + 2)
MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
39
Which of the following is a factor of 12 x 2  x  6?
a.
b.
 3x  2 
 4 x  3
 4 x  3
c.
d. not factorable
Solution: c. (4x + 3)
MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
40
Which of the following is a factor of 3x3  18 x 2  48 x ?
a.  x  4 
b.  x  2 
c.  x  8 
d.  x  2 
Solution: b. (x + 2)
MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
41
Write in simplest radical form.
180x9 y 8
Solution:
6 x4 y 4 5x
MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
42
Write in simplest radical form.
49x 4 y 2
Solution: 7x2y
MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
43
Write in simplest radical form.
2

2 3

Solution: 2  3 2
MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
44
Write in simplest radical form.
3 32  7 50
Solution: 23 2
MATHEMATICS for COLLEGE READINESS
Algebraic Expressions
45
Perform the indicated operations and reduce
answers to lowest terms. m  n
m 2  mn
m n
2
a.
m
2

m  2mn  n 2
 m  n  m  n 
mn
b.
mm  n
c. m
1
d. m
MATHEMATICS for COLLEGE READINESS
Solution: d
Quadratic Equations
46
Solve algebraically.
25 x 2  144  0
12 12
,
Solution: x 
5 5
MATHEMATICS for COLLEGE READINESS
Quadratic Equations
47
Solve algebraically.
6 x 2  150
Solution: x   5,5
MATHEMATICS for COLLEGE READINESS
Quadratic Equations
48
Solve algebraically.
6
6x  5 
x
2 3
,
Solution: x 
3 2
MATHEMATICS for COLLEGE READINESS
Quadratic Equations
49
Solve algebraically.
x2  x  4
1  17
Solution: x 
2
MATHEMATICS for COLLEGE READINESS
Quadratic Equations
50
Solve algebraically.
3x 2  2  5 x
Solution:
x
1
,2
3
MATHEMATICS for COLLEGE READINESS
Rational Expressions
51
State the restricted values:
Solution: x ≠ –3, 2
MATHEMATICS for COLLEGE READINESS
 x  1
1
2
x2  x  6
Rational Expressions
52
Over the set of real numbers, state where the function is
undefined:
x2  4
x2  9
Solution:
The expression is undefined over the set of real numbers
where -2<x<2 or x = -3 or 3.
MATHEMATICS for COLLEGE READINESS
Rational Expressions
53
x2  4 x2  5x  6
Find the product:
3 x  6 4 x  12
 x  2 2
Solution:
12
MATHEMATICS for COLLEGE READINESS
Rational Expressions
54
Calculate and write the answer in scientific notation.
85 10  0.03 10 
 0.5 10  3.2 10 
4
1
a. 1.59375 1011
b. 1.59375 1011
c. 1.59375 101
d. 1.59375 109
Solution:
a
MATHEMATICS for COLLEGE READINESS
2
8
Rational Expressions
55
x2  y 2
,
Find the area of a rectangle if the length is
y
x y
and the width is
.
x y
Solution:
( x  y)2
y
MATHEMATICS for COLLEGE READINESS
Rational Expressions
56
5x
5

Add and simplify x 2  1 x 2  1 .
5
Solution:
x 1
MATHEMATICS for COLLEGE READINESS
Rational Expressions
57
3a
Simplify 2b .
2b
5a
Solution:
3
5
MATHEMATICS for COLLEGE READINESS
Rational Expressions
58
3
4
Simplify x
.
x 1
x2
3x  4 x 2
Solution:
x 1
MATHEMATICS for COLLEGE READINESS
Rational Expressions
59
x3  y 3 x 2  y 2

Simplify:
2
x
2x
2 x 2  2 xy  2 y 2
Solution:
x 2  xy
MATHEMATICS for COLLEGE READINESS
Rational Expressions
60
Simplify:
x x  y
6
x y
3
18
a.
x  x  y  x  y 
b. 2x
c.
2x  x  y 
x y
6x
d.
3
MATHEMATICS for COLLEGE READINESS
Solution: b
Rational Expressions
61
Simplify:
3
a. 2  x  1
3 x2  x  2
2
2
x

4
x

   1
3
b.
2  x  1
c.
3 x  2
2  x  2  x  1
3
d.
2
MATHEMATICS for COLLEGE READINESS
Solution: a
Rational Expressions
62
5v  5
v2 1
Simplify: v  2  v 2  4v  4
 
a.
b.
5
 v  2  v  1
5  v  1 v  2 
 v  1 v  1
Solution: d
MATHEMATICS for COLLEGE READINESS
c.
d.
5  v  5  v  2 
 v  1 v  1
5v  2
 v  1
Rational Expressions
63
What values for x are not allowed?
3x  4 x  5
x2  2 x  8
5
a. 0,
4
5
c. 4,0, , 2
4
b. 4,2
d. 4, , 2
Solution: b
MATHEMATICS for COLLEGE READINESS
5
4
Miscellaneous
64
Which of the following sets of numbers could NOT
be the measures of the sides of a right triangle?
a. 3,7, 10
c. 6,8,10
b. 8,5 3, 11
d. 10,24,26
Solution: a
MATHEMATICS for COLLEGE READINESS
Miscellaneous
65
Find the length of the hypotenuse of the right
triangle whose legs are 5 and 11.
Solution: 6
MATHEMATICS for COLLEGE READINESS
Miscellaneous
66
Find the distance between (-3,5) and (-7,8).
Solution: 5
MATHEMATICS for COLLEGE READINESS
Miscellaneous
67
Find the coordinates of the point midway between
(-5,2) and (3,-10).
a. (-4,6)
b. (-1,-4)
c. (1,6)
d. (-4,-1)
Solution: b. (-1, -4)
MATHEMATICS for COLLEGE READINESS
Miscellaneous
68
Find y in the diagram below.
a. 24
b. 18
c. 4
d. 8
9
6
12
Solution: a
MATHEMATICS for COLLEGE READINESS
y
Miscellaneous
69
Find y in the diagram below.
12
8
y
Solution: 7.5
MATHEMATICS for COLLEGE READINESS
15
Variation
70
Find an equation of variation where y varies
inversely as x, and y  10 when x  3.5
a.
b.
c.
d.
y  35 x
3
y
x
y  6.5 x
35
y
x
Solution: d
MATHEMATICS for COLLEGE READINESS
Variation
71
Y varies directly as the cube root of x. Y is 30 when x
is 1. Find Y when x is 27.
a.
b.
c.
d.
10
90
15
30
Solution: b. 90
MATHEMATICS for COLLEGE READINESS
Variation
72
The number of hours (H) it takes to do the job in inversely
proportional to the number of people (P) working on it. It
takes 4 people 12 hours to do the job. How long would it take
6 people to do the job?
a.
b.
c.
d.
18 hours
10 hours
8.5 hours
8 hours
Solution: d
MATHEMATICS for COLLEGE READINESS
Content Guide
Equations and Modeling
Function Theory
Expressions with Exponents
Exponential and Logarithmic Functions
Simplifying Expressions
Algebraic Expressions
Quadratic Equations
Rational Expressions
Miscellaneous
Variation
MATHEMATICS for COLLEGE READINESS
Slides 1 – 15
Slides 16 – 21
Slides 22 – 30
Slides 31 – 32
Slides 33 – 36
Slides 37 – 45
Slides 45 – 50
Slides 51 – 63
Slides 64 – 69
Slides 70 – 72