review for college mathematics

A SELECTION OF PROBLEMS INTENDED TO MAINTAIN STUDENTS ESSENTIAL ALGEBRA

SKILLS

INSTRUCTOR COPY

Review for College Mathematics

2

MATHEMATICS for COLLEGE READINESS

Equations and Modeling

A moving company charges $46 to move a certain machine 10 miles and $58 to move the same machine 30 miles.

a) Find an equation that defines this relationship if it is linear.b) What will it cost to move the machine 25 miles?c) What is the minimum charge for moving the machine?d) What is the rate for each mile the machine is moved?

Solution:a)b)c)d)

C = 0.6 m + 14$55$40$0.60 per mile

3



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) b)c)d)

C = 4x + 210, where x = number of units produced.$61

0$210$4 per unit

4



If x2 + 2x = 3, demonstrate algebraically which of the following is a possible value for x.a) – 3b) –2c) –1d) 0e) 3

Solution:

a) – 3

5



How many solutions are there to the systems of equations? Solve algebraically.

Solution:Infinite: the lines coincide so the solution is

x 2y6y3x

6 1

, : 2 6x y x y

6



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)

2 3 40n n n =5

7



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.

8



Solve the system of equations algebraically.

Solution:empty set: The lines are parallel.

3x4y2y3x

4 1

6

9



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:

22

2

p lw

pw l

10



Solve for a:

Solution:

1a

31b

1 3bab

11



Solve for x:

Solution:

1x 1y1a

ayxy a

12



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

13



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.

14



Solve algebraically: x2 – 5 = 3x

Solutions:

3 292

x

15


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.

16



Find the equation of a line with slope of 0 through (6, 10).

Solution:

y = 10

17

Function Theory


Which of the following are functions?

a. x3 y = 12

b. x2 + y2 = 20

c. x3 + y = 15

d. x (x+2) = 5 + y

3 42x y x e.

f. y(y + 2) = 4 + x

g. x = 4

h. y = 5

Solution: c, d, e, h

18

Function Theory


If f(x) = x2 – 5, find f(b) – f( c) .

Solution: b2 – c2

19

Function Theory


If f(x) = 2x2 – x + 3, find f(x + h) .

Solution:

2 22 4 2 3x xh h x h

20

Function Theory


If f(x) = x2 – 4, find f(a + 5) .

Solution:

a2 +10a+21

21

Function Theory


If f(x) = x2 – 3 x + 5, find f(x + h) – f(x) .

Solution:2xh + h2 – 3h

22

Function Theory


If f(x) = 3x4 – 4x3 -2x2 +x - 5, find f(-x).

Solution:

2xh + h2 – 3h

23

Expressions with Exponents


Simplify:

Solution:

10 2

1

x yy

3

1

y

24


Simplify. Give the answer with positive exponents only:


3 0

4

2 3x x

Solution: 4

8x

25


Simplify. Give the answer with positive exponents only:


2 3 4

1 2 2

6 .4x y zx y z

5

6

3 .2yx z

Solution:

26


Multiply

a.

b. c.

d.

e.

Solution:


8r2s 3r5s4

11r7s5

24r7s5

7 5d. 24r s

5r7s5

5r10s4

24r10s4

27


Solution:


1/2 2/5 1/49r s r s

9/10 5/4 9r s

Multiply

28


Solution:


1/2 5/4 016r s r s

3/4 16r s

Multiply

29


If the area of a circle if what is the area of a circle if the radius is 3a?


2 ,r

Solution: 29A a

30


If the volume of a sphere is what is the volume of a sphere with radius 3a?


3 ,43r

Solution: 336V a

31


If the volume of a cylinder is find the volume of a cylinder with radius 3x and a height that is twice the radius.


2 ,r h

354V aSolution:

32


If , then x =

a. 3 + ln t

b.

Exponential and Logarithmic Functions

3xt e

t 3e

t3

c.

d. ln ( t – 3 )

e. ln t – 3

Solution: e. ln t – 3

33


If 7x = 3, then x =

a.

b.

37

73

log3 7c.

d.

e.

7log 3

log1073

Solution:

3c. log 7

Exponential and Logarithmic Functions

34


Simplifying Expressions

Simplify

a.b.c.d.

3 2 3 4x x x x

2

2

2

2

21 4

27 4

10 20

10 10

x x

x x

x x

x x

Solution:

a

35



Simplify

a.b.c.d.

22 22 2a b a b

2 2

2 2

2 2

2 2

3

5

4 5

4 3

a b

a b

a ab b

a ab b

Solution:

c

36



Simplify

230 7 4 5k k k

Solution:

5k2 + 250k -16

37



Evaluate the following expression using x = -3.1 and y = 4.23

4x-2 + y3

Solution:

76.103

38


Factor completely using integer coefficients.

Algebraic Expressions

4 3 2 312 48m n m n

Solution: 12 m2n3 (m2 + 4)

39


Factor completely using integer coefficients.


212 8 9 6m m m

Solution:(4m – 3) (3m + 2)

40



Which of the following is a factor of

a.b.c.d. not factorable

212 6?x x

3 2

4 3

4 3

x

x

x

Solution:c. (4x + 3)

41



Which of the following is a factor of

a.b.c.d.

3 23 18 48 ?x x x

4

2

8

2

x

x

x

x

Solution: b. (x + 2)

42



Write in simplest radical form.

9 8180x y

Solution: 4 46 5x y x

43




4 249x y

Solution:7x2y

44




2 2 3

Solution: 2 3 2

45




3 32 7 50

Solution: 23 2

46



Perform the indicated operations and reduce answers to lowest terms.

a.

b.

c.

d. 2

2 2 22m n m mnm n m mn n

1

mm n m n

m nm m n

m

m

Solution: d

47


Solve algebraically.

Quadratic Equations

225 144 0x

Solution:12 12,5 5

x

48



Quadratic Equations

26 150x

Solution: 5,5x

49



Quadratic Equations

66 5xx

Solution: 2 3,3 2

x

50



Quadratic Equations

2 4x x

Solution: 1 172

x

51



Quadratic Equations

23 2 5x x

Solution:1, 2

3x

52

Rational Expressions


State the restricted values:

Solution:

12

2

16

xx x

x ≠ –3, 2

53


Over the set of real numbers, state where the function is undefined:


2

2

49

xx

Solution: The expression is undefined over the set of real numbers where -2<x<2 or x = -3 or 3.

54


Find the product:


2 24 5 63 6 4 12x x xx x

Solution: 2212x

55


Calculate and write the answer in scientific notation.

a.b.c.d.


4 2

1 8

85 10 0.03 10

0.5 10 3.2 10

11

11

1

9

1.59375 10

1.59375 10

1.59375 10

1.59375 10

a Solution:

56


Find the area of a rectangle if the length isand the width is


2 2

,x yy

.x yx y

Solution:2( )x y

y

57


Add and simplify


2 2

5 5 .1 1x

x x

Solution: 51x

58


Simplify


32 .25

abba

Solution: 35

59


Simplify


2

3 4.1

xxx

Solution:23 4

1x xx

60


Simplify:


3 3 2 2

2 2x y x yx x

Solution:2 2

2

2 2 2x xy yx xy

61


Simplify:

a.

b.

c.

d.


63

x x yx y

63x

18

x x y x y

2x

2x x yx y

Solution: b

62


Simplify:

a.

b.

c.

d.


2

2

3 2

2 4 1

x x

x x

3

2 1x

3

2 1x

3 22 2 1

xx x

32

Solution: a

63


Simplify:

a. c.

b. d.


2

2

5 5 12 4 4

v vv v v

5

2 1v v

5 1 21 1

v vv v

5 5 21 1

v vv v

5 21

vv

Solution: d

64


What values for x are not allowed?

a. c.

b. d.


2

3 4 52 8

x xx x

50,4

4,2

54,0, ,24

54, ,24

Solution: b

65


Miscellaneous

Which of the following sets of numbers could NOT be the measures of the sides of a right triangle?

a. c.

b. d.

3,7, 10

8,5 3, 11

6,8,10

10,24,26

Solution: a

66


Miscellaneous

Find the length of the hypotenuse of the right triangle whose legs are 5 and 11.

Solution: 6

67


Miscellaneous

Find the distance between (-3,5) and (-7,8).

Solution: 5

68

Miscellaneous


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)

69


Miscellaneous


Find y in the diagram below.

a. 24b. 18c. 4d. 8

Solution:

12

69

y

a

70


Miscellaneous


Find y in the diagram below.

Solution:

y

812

15

7.5

71


c.

d.

Variation

Find an equation of variation where y varies inversely as and when

a.

b. ,x 10y 3.5x

35y x3yx

6.5y x35yx

Solution: d

72


Variation

Y varies directly as the cube root of x. Y is 30 when x is 1. Find Y when x is 27.

a. 10b. 90c. 15d. 30

Solution:

b. 90

73


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. 18 hoursb. 10 hoursc. 8.5 hoursd. 8 hours

Variation

Solution: d


Equations and Modeling Slides 1 – 15Function Theory Slides 16 – 21Expressions with Exponents Slides 22 – 30Exponential and Logarithmic Functions Slides 31 – 32Simplifying Expressions Slides 33 – 36Algebraic Expressions Slides 37 – 45Quadratic Equations Slides 45 – 50Rational Expressions Slides 51 – 63Miscellaneous Slides 64 – 69VariationSlides 70 – 72

Content Guide

review for college mathematics

Documents