College Algebra Study Guide › media › 11179 › college-math-aims.pdf3 October 2012 College Level Math B. Solving Absolute value Inequalities: Solve and Graph. 1. 4 1 6x d 3. 5

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1 August 2012 College Level Math

Accuplacer College Level Math Study Guide The following sample questions are similar to the format and content of questions on the Accuplacer College Level

Math test. Reviewing these samples will give you a good idea of how the test works and just what mathematical

topics you may wish to review before taking the test itself. Our purposes in providing you with this information are to

aid your memory and to help you do your best.

I. Factoring and expanding polynomials

Factor the following polynomials: Expand the following:

1.3 2 2 3 215 45 60a b a b a b 9. 1 1 3x x x

2. 3 3 2 2 3 2 27 21 10 30x y x y x y x y 10. 2

2 3x y

3. 4 4 3 2 26 6 8 8x y x y xy 11. 3 3 6 6x x

4. 2 22 7 6x xy y 12. 2

2 2 3x x

5. 4 2 6y y 13. 5

1x

6. 3 37 56x y 14. 6

1x

7. 4 481 16r s

8. 2

2 1x y x y

II. Simplification of Rational Algebraic Expressions

Simplify the following. Assume all variables are larger than zero.

1. 2 03 5 4 4 4. 2 18 5 32 7 162

2. 9 3 5 8 2 27

3. 4

81

x5.

2

6 18 12 16

3 2 8 4 12

x x

x x x

III. Solving Equations

A. Solving Linear Equations

1. 3 2 1 10x x 3. 22 6y y y

2. 12 7

x x 4. 2 1 3 3 1x x x

B. Solving Quadratic & Polynomial Equations

1. 8 2

03 3

y y

5. 2 1 0t t

2. 3 22 4 30 0x x x 6.

33 24x

3. 327 1x 7.

2 21 25x x

4. 3 6 9 22x x x 8. 25 1y y

2 October 2012 College Level Math

C. Solving Rational Equations

1. 1 2

01 1y y

4.

2

11 2 1

25 5 5x x x

2. 2

2 3 12

3 3 9x x x

5.

2

1 6

5a a

3. 2

1 2 5

6 3 3 18

x

x x x x

6.

2

1 1

3 3

x

x x x x

D. Solving Absolute Value Equations

1. 5 2 1 8z 4. 1 3 1

2 4 4x

2. 5 7 2x 5. 1 7y y

3. 5 1 2x

E. Solving Exponential Equations

1. 10 1000x 4. 2 1

3 93

x x

2. 3 510 100x

3. 1 1

28

x 5. 2 2 1

2 48

x x

F. Solving Logarithmic Equations

1. 2 2log 5 log 1 5x x 4. ln ln 2 1 0x x

2. 3 32log 1 log 4x x 5. ln ln 2 ln3x x

3. 2 2log 1 log 1 3x x 6. 2 13 4x x

G. Solving Radical Equations

1. 4 2 1 2 0y 4. 2 9 1 0x x

2. 2 1 5 8x 5. 3 3 2 4 6x

3. 5 1 2 1 0x x 6. 24 7 2w

IV. Solving Inequalities

Solve the following inequalities and express the answer graphically and using interval notation.

A. Solving Linear Inequalities

1. 3

4 25

x 3. 3 2 6 3 14x x

2. 3 3 5 1x x 4. 2 3 10 5x


B. Solving Absolute value Inequalities: Solve and Graph.

1. 4 1 6x 3. 5

53

x

2. 4 3 2 9x 4. 5 2 15x

C. Solving Quadratic or Rational Inequalities

1. 23 11 4 0x x 3.

20

3

x

x

2. 26 5 4x x 4.

1 30

2 7

x x

x

V. Lines & Regions

1. Find the x and y-intercepts, the slope, and graph 6x + 5y = 30.

2. Find the x and y-intercepts, the slope, and graph x = 3.

3. Find the x and y-intercepts, the slope, and graph y = -4.

4. Write in slope-intercept form the line that passes through the points (4, 6) and (-4, 2).

5. Write in slope-intercept form the line perpendicular to the graph of 4x - y = -1 and containing the point (2, 3).

6. Graph the solution set of x - y ≥ 2.

7. Graph the solution set of -x + 3y < -6.

VI. Graphing Relations, Domain & Range

For each relation, state if it is a function, state the domain & range, and graph it.

1. 2y x 6. 2 2x y

2. 2y x 7. 2 8 6y x x

3. 1

2

xy

x

8. y x

4. 1 3f x x 9. 3xy

5. 2

2 5

9

xf x

x

10.

2

2

6

3 2 1

xh x

x x

VII. Exponents and Radicals

Simplify. Assume all variables are >0. Rationalize the denominators when needed.

1. 3 38x 6.

26 2

3 8

54

9

a b

a b

2. 5 147 4 48 7.

3 3

3 2 2

27

2

a

a b

3. 5 15 3 8. 2

5 3

4.

32 4

3 3

5

3

x y

x

9. 3

x

x

5.

4

39

40x

y


VIII. Complex Numbers

Perform the indicated operation and simplify.

1. 16 4 9 5. 2

4 3i

2. 16 9 6. 25i

3. 16

9

7.

3 2

4 5

i

i

4. 4 3 4 3i i

IX. Exponential Functions and Logarithms

1. Graph: 3 1xf x 6. Solve: log 9 2x

2. Graph: 12xg x 7. Graph: 3logh x x

3. Express2 1

864

in logarithmic form 8. Use the properties of logarithms to expand as

4. Express 5log 25 2 in exponential form much as possible: 4

3log

y

5. Solve: 2log 4x 9. How long will it take $850 to be worth $1000 if

it is invested at 12% interest compounded

quarterly?

X. Systems of Equations & Matrices

1. Solve the system:2 3 7

6 1

x y

x y

4. Multiply:

1 1 1 0 2 1

0 2 0 1 2 0

2 1 3 0 0 1

2. Solve the system:

2 2 3

2 3 6 2

0

x y z

x y z

x y z

5. Find the determinant:1 2

3 1

3. Perform the indicated operation: 6. Find the Inverse:1 2

1 2

13

3 1 22 3

1 2 1 6

XI. Story Problems

1. Sam made $10 more than twice what Pete earned in one month. If together they earned $760, how much

did each earn that month?

2. A woman burns up three times as many calories running as she does when walking the same distance. If

she runs 2 miles and walks 5 miles to burn up a total of 770 calories, how many calories does she burn up

while running 1 mile?

3. A pole is standing in a small lake. If one-sixth of the

length of the pole is in the sand at the bottom of the

lake, 25 ft. are in the water, and two-thirds of the total

length is in the air above the water, what is the length

of the pole?

Water Line

Sand


XII. Conic Sections

1. Graph the following, and find the center, 2. Identify the conic section and put it into standard

foci, and asymptotes if possible. form.

a) 2 2( 2) 16x y a)

2 24 12 0x x y

b)

2 2( 1) ( 2)1

16 9

x y b)

2 29 18 16 64 71x x y y

c)

2 2( 1) ( 2)1

16 9

x y c)

2 29 18 16 64 199x x y y

d) 2( 2) 4x y d)

2 4 0x y x

XIII. Sequence and Series

1. Write out the first four terms of the sequence whose general term is 3 2na n

2. Write out the first four terms of the sequence whose general term is 2 1na n

3. Write out the first four terms of the sequence whose general term is 2 1n

na

4. Find the general term for the following sequence: 2,5,8,11,14,17....

5. Find the general term for the following sequence: 1 12 4

4,2,1, , ,....

6. Find the sum:

6

0

2 1k

k

7. Expand the following:

44

0

4k k

k

x yk

XIV. Functions

Let ( ) 2 9f x x and 2( ) 16g x x . Find the following.

1. ( 3) (2)f g 5. ( )( 2)g f

2. (5) (4)f g 6. ( ( ))f g x

3. ( 1) ( 2)f g 7. 1(2)f

4. (5)

(5)f

g 8. 1(3)f f

XV. Fundamental Counting Rule, Factorials, Permutations, & Combinations

1. Evaluate:

8!

3! 8 3 !

2. A particular new car model is available with five choices of color, three choices of transmission, four types

of interior, and two types of engines. How many different variations of this model car are possible?

3. In a horse race, how many different finishes among the first three places are possible for a ten-horse race?

4. How many ways can a three-Person subcommittee be selected from a committee of seven people? How

many ways can a president, vice president, and secretary be chosen from a committee of seven people.


XVI. Trigonometry

1. Graph the following through on period: ( ) sinf x x

2. Graph the following through on period: ( ) cos(2 )g x x

3. A man whose eye level is 6 feet above the ground stands 40 feet from a building. The angle of elevation

from eye level to the top of the building is 72 . How tall is the building?

4. A man standing at the top of a 65m lighthouse observes two boats. Using the data given in the picture,

determine the distance between the two boats.


Answers I. Factoring and Expanding Polynomials

When factoring, there are three steps to keep in mind.

1. Always factor out the Greatest Common Factor

2. Factor what is left

3. If there are four terms, consider factoring by grouping.

Answers:

1. 2 215 ( 3 4)a b ab b

2. 2 (7 10)( 3)x y y xy

3 3 2 2 3 2 2

2 2

2 2

2

7 21 10 30

(7 21 10 30)

(7 21 ) ( 10 30)

7 ( 3) 10( 3)

x y x y x y x y

x y xy y xy

x y xy y xy

x y y xy xy

3. 3 2 22(3 4)( 1)x y xy

4. (2 3 )( 2 )x y x y

5. 2 2( 2)( 3)y y

4 2

2

6

6

( 2)( 3)

y y

u u

u u

6. 2 27( 2 )( 2 4 )x y x xy y

7. 2 2(3 2 )(3 2 )(9 4 )r s r s r s

8. 2( 1)x y Hint: Let u=x+y

9. 3 23 3x x x

10. 2 24 12 9x xy y

11. 23 2 3 2x

12. 4 3 24 10 12 9x x x x

13. 5 4 3 25 10 10 5 1x x x x x

14. 6 5 4 3 26 15 20 15 6 1x x x x x x

Since there are 4 terms, we consider factoring by grouping.

First, take out the Greatest Common Factor.

When you factor by grouping, be careful of the minus sign

between the two middle terms.

When a problem looks slightly odd, we can make it appear more natural to us

by using substitution (a procedure needed for calculus). Let 2u y Factor

the expression with u’s. Then, substitute the 2y back in place of the u’s. If

you can factor more, proceed. Otherwise, you are done.

Formula for factoring the sum of two cubes:

3 3 2 2( )( )a b a b a ab b

The difference of two cubes is:

3 3 2 2( )( )a b a b a ab b

When doing problems 13 and 14, you may want to use

Pascal’s Triangle

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1


II. Simplification of Rational Algebraic Expressions 1. 13

2. 38

3. 2

9

x

4. 49 2

5. 6

2x

III. Solving Equations

A. Solving Linear Equations

1. 5x 2. 14

5x or

42

5 3. 3y 4. 1x

B. Solving Quadratic & Polynomial Equations

1. 8 2

,3 3

y

2. 0, 3,5x

3. 1 1 3

,3 6 6

ix

4. 10, 4x

5. 1 3

2 2

it

6. 2, 1 3x i

7. 3, 4x

8. 1 21

10y

C. Solving Rational Equations

1. 1

3y

1 20

1 1

1 2( 1)( 1) 0( 1)( 1)

1 1

1 2( 1)( 1) ( 1)( 1) 0

1 1

( 1) 2( 1) 0

3 1 0

y y

y y y yy y

y y y yy y

y y

y

2. Working the problem, we get 3x . However, 3 causes the denominators to be zero in the original

equation. Hence, this problem has no solution.

If you have 4 , you can write 4 as a product of primes (2 2) . In square

roots, it takes two of the same thing on the inside to get one thing on the

outside: 4 2 2 2

Solving Quadratics or Polynomials:

1. Try to factor

2. If factoring is not possible, use the quadratic formula

2 4

2

b b acx

a

where

2 0ax bx c

Note: 1i

Example: 12 12 2 2 3 2 3i i i

Solving Rational Equations:

1. Find the lowest common denominator for

all fractions in the equation

2. Multiply both sides of the equation by the

lowest common denominator

3. Simplify and solve for the given variable

4. Check answers to make sure that they

do not cause zero to occur in the

denominators of the original equation.


3. 15

4x

4. 2x

5. 1, 5a

6. 2,1x

D. Solving Absolute Value Equations

1. 2z or 7z

5 2 1 8

5 2 9

z

z

5 2 9z or 5 2 9z

2 4z or 2 14z 2. 0x or 10x 3. No solution. An absolute value

cannot equal a negative number

4. 2x or 1x

5. 3y

1 7y y

1 7y y or 1 (7 )y y

0 8 or 2 6y

No solution or 3y

Hence, 3y is the only solution

E. Solving Exponential Equations

1. 3x

3

10 1000

10 10

3

x

x

x

2. 1x

3. 4x

4. 1, 1x

5. 1, 3x

F. Solving Logarithmic Equations

1. 2

3x

2 2log ( 5) log (1 5 )

5 1 5

6 4

x x

x x

x

2. 1,1x

3 3

2

3 3

2

2

2log ( 1) log (4 )

log ( 1) log (4 )

( 1) 4

2 1 0

x x

x x

x x

x x

Solving Absolute Value Equations:

1. Isolate the Absolute value on one side of the equation

and everything else on the other side

2. Remember that 2x means that the object inside

the absolute value has a distance of 2 away from zero.

The only numbers with a distance of 2 away from zero

are 2 and -2. Hence, 2x or 2x . Use the

same thought process for solving other absolute value

equations

Note: An absolute value cannot equal a negative value.

2x does not make any sense.

Note: Always check your answers!!

Some properties you will need to be familiar with.

If r sa a , then r s

If r ra b , then a b

Properties of logarithms to be familiar with:

If log logb bM N , then M N

If logb x y , then this equation can be rewritten in

exponential form as yb x

log ( ) log logb b bM N M N

log log logb b b

MM N

N

log logr

b bM r M

Always check your answer!! Bases and arguments of

logarithms cannot be negative


3. 3x is the only solution since -3 causes the argument of a logarithm to be negative

4. 1

2x is the only solution since -1 causes the argument of a logarithm to be negative

5. 1x is the only solution since -3 causes the argument of a logarithm to be negative

6. ln 4

2ln 3 ln 4x

2 13 4

2 ln 3 ( 1) ln 4

2 ln 3 ln 4 ln 4

(2ln 3 ln 4) ln 4

x x

x x

x x

x

G. Solving Radical Equations

1. 5

8y

4 2 1 2 0

12 1

2

12 1

4

52

4

y

y

y

y

2. 4x

3. 5x

2 2

5 1 2 1 0

5 1 2 1

5 1 2 1

5 1 4 1

x x

x x

x x

x x

4. No solution. 4x does not work in the original equation.

5. 2x

6. 3, 3w

IV. Solving Inequalities

A. Linear

1. 3

4 25

x

3

65

x

10x

Interval Notation: , 10

Solving Equations with radicals:

1. Isolate the radical on one side of the equation and everything

else on the other side

2. If it is a square root, then square both sides. If it is a cube root,

then cube both sides, etc…

3. Solve for the given variable and check your answer

Note: A radical with an even index such as 64, , ,…

cannot have a negative argument (The square root can but

you must use complex numbers).

When solving linear inequalities, you use the same steps as

solving an equation. The difference is when you multiply or

divide both sides by a negative number, you must change

the direction of the inequality.

For example:

5 3

1 5 1 3

5 3


2. 7x Interval Notation: ,7

3. 4x Internal Notation: 4,

4. 4 5x Internal Notation: 4,5

B. Absolute Value

1.

4 1 6

6 4 1 6

7 4 5

7 5

4 4

x

x

x

x

Interval: 7 5

,4 4

2. 1x or 5

2x

Interval: 5

, 1,2

3. 20x or 10x

Interval: , 20 10,

4. 5 10x

Interval: 5,10

Think of the inequality sign as an alligator. If the alligator is

facing away from the absolute value sign such as, 5x ,

then one can remove the absolute value and write

5 5x . This expression indicates that x cannot be

farther than 5 units away from zero.

If the alligator faces the absolute value such as, 5x ,

then one can remove the absolute value and write 5x

or 5x . These expressions express that x cannot be

less than 5 units away from zero.


C. Quadratic or Rational

1.

23 11 4 0

3 1 4 0

x x

x x

1

3x and 4x make the above factors zero.

Answer: 1

,43

3 6x y

2. 4 1

, ,3 2

3. 2,3

4. 7

, 1,32

V. Lines and Regions

1. x – intercept: (5,0)

y – intercept: (0,6)

slope: 6

5

2. x – intercept: (3,0)

y – intercept: None

slope: None

Steps to solving quadratic or rational

inequalities.

1. Zero should be on one side of the

inequalities while everything else is on

the other side.

2. Factor

3. Set the factors equal to zero and

solve.

4. Draw a chart. You should have a

number line and lines dividing regions

on the numbers that make the factors

zero. Write the factors in on the side.

5. In each region, pick a number and

substitute it in for x in each factor.

Record the sign in that region.

6. In our example, 3x+1 is negative in

the first region when we substitute a

number such as -2 in for x. Moreover,

3x+1 will be negative everywhere in

the first region. Likewise, x-4 will be

negative throughout the whole first

region. If x is a number in the first

region, then both factors will be

negative. Since a negative times a

negative number is positive, x in the

first region is not a solution. Continue

with step 5 until you find a region that

satisfies the inequality.

7. Especially with rational expression,

check that your endpoints do not

make the original inequality

undefined.

3x + 1

x - 4

4

- + +

- - -


3. x – intercept: None

y – intercept: (0,-4)

slope: 0

4. 1

42

y x

5. 1 1

34 2

y x

6. 2x y

7. 3 6x y


VI. Graphing Relations

1. 2y x

Domain: 2,

Range: 0,

2. 2y x

Domain: 0,

Range: 2,

3. 1

2

xy

x

Domain: All Real Numbers except -2

Range: ,1 1,

4. 1 3f x x

Domain: ,

Range: ,3


5. 2

2 5

9

xf x

x

Domain: All Real Number except 3

Range: All Real Numbers

6. 2 2x y

Domain: 2,

Range: ,

7. 2 8 6y x x

Domain: ,

Range: 22,

8. y x

Domain: ,0

Range: 0,


9. 3xy

Domain: ,

Range: 0,

10. 2

2

6

3 2 1

xh x

x x

Domain: All Real Numbers except: 1

,13

Range: 2, ,0

VII. Exponents and Radicals

1. 2x

2. 5 147 4 48 35 3 16 3 19 3

3. 5 3 15

4.

7

4

x

y

5.

3

3

2 5x x

y

6.

2 26 2 6 12

3 8 3 6

54 6

9 36

a b a b

a b a b

7.

3 3 3 3 3

33 3 32 2 2 2 2 2

27 3 3 4 3 4 3 4

2 242 2 2

a a a ab a ab ab

ab baba b a b a b

8. 5 3

9. 3 3

93 3

x x x x x

xx x


VIII. Complex Numbers

1. 16 4 9 4 12 8i i i

2. 216 9 4 3 12 12i i i

3.

2

2

16 4 4 3 12 12 4

3 3 3 9 9 39

i i i i

i i i i

4. 24 3 4 3 16 9 16 9 25i i i

5. 2 24 3 4 3 4 3 16 24 9 16 24 9 7 24i i i i i i i

6. 12 1225 24 2 1i i i i i i i

7.

2

2

3 2 4 5 12 23 10 12 23 10 2 23

4 5 4 5 16 25 16 25 41 41

i i i i i i

i i i

IX. Exponential Functions and Logarithms

1. 3 1xf x

2. 12xg x

3. 8

1log 2

64

4. 25 25

5.

2

4

log 4

2

16

x

x

x


6. x=3; -3 is not a solution because bases are not allowed to be negative

7. 3logh x x

8. 4 4 4

3log log 3 log y

y

9. 1

ntr

A Pn

where

4

4

4

0.121000 850 1

4

20 0.121

17 4

20 0.12log log 1

17 4

20 0.12log 4 log 1

17 4

20log

17

0.124log 1

4

t

t

t

t

t

A = Money ended with

P = Principle started with

r = Yearly interest rate

n = Number of compounds per year

t = Number of years


X. Systems of Equations

1. 1 3

,2,2 4

2.

3 2

5 5

8 2

5 5

k

k

k

for k Natural numbers

3. 5 8

5 14

4.

1 0 2

2 4 0

1 2 5

5. 5 6.

1 1

2 2

1 1

4 4

XI. Story Problems

1. Let x = the money Pete earns 2 10 760x x Pete earns $250

2 10x the money Sam earns Sam earns $510

2. x = burned calories walking 2 3 5 770

70

x x

x

3x = burned calories running Answer: 210 calories

3. x = length of pole 2 1

253 6

x x x Answer: 150 feet

XII. Conic Sections

1. a) 2 22 16x y

Center: (2,0)

Radius: 4


b)

2 21 2

116 9

x y

Center: (-1,2)

Foci: 1 7,2

c)

2 21 2

116 9

x y

Center: (-1,2)

Foci: (-6,2), (4,2)

Asymptotes:

3 3

4 4

3 5

4 4

y x

y x

d) 2

2 4x y

Vertex: (2,4)

Foci: 15

2,4

Directrix: 17

4y

2. a) Circle 2 22 16x y

b) Ellipse

2 21 2

116 9

x y

c) Hyperbola

2 21 2

116 9

x y

d) Parabola 2

2 4y x


XIII. Sequence and Series

1. 1, 4, 7, 10

2. 0, 3, 8, 15

3. 3, 5, 9, 17

4. 3 1na n

5.

131

4 22

nn

na

6. 6

0

2 1 1 1 3 5 7 9 11 35k

k

7.

44 4 3 2 2 3 4

0

44 6 4k k

k

x y y xy x y x y xk

XIV. Functions

1. 3 2 3 12 15f g

2. 5 4 19 0 19f g

3. 1 2 7 12 84f g

4.

5 19 19

5 9 9

f

g

5. 2 2 5 9g f g f g

6. 2 2 216 2 16 9 2 41f g x f x x x

7. 1 9;

2

xf x

1 2 9 72

2 2f

8. 3


XV. Fundamental Counting Rule, Permutations, & Combinations

1. 56

2. 120

3. 720

4. Committee 35

Elected 210

XVI. Trigonometry

1. sinf x x

2. cos 2g x x

3. 6 40tan72 129.1x

4. Distance between the boats 65tan 42 65tan35 50 11.59meters

Documents

College Algebra Study Guide › media › 11179 › college-math-aims.pdf3 October 2012 College Level Math B. Solving Absolute value Inequalities: Solve and Graph. 1. 4 1 6x d 3. 5