Summary College Algebra Study Guide: Functions, Equations, Graphs & Problem Solving

Student Success Center
College Level Math Study Guide for the
ACCUPLACER (CPT)

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. 15a b − 45a b − 60a b
3 2 2 3 2
9. (x + 1)(x − 1)(x − 3)
2. 7 x 3 y 3 + 21x 2 y 2 − 10 x 3 y 2 − 30 x 2 y 10. (2 x + 3y )2
3. 6x 4 y 4 − 6 x 3 y 2 + 8xy 2 − 8 11. ( 3x + 3 )( 6x − 6 )
4. 2x 2 − 7 xy + 6 y 2
12. (x 2
− 2x + 3 )
2

y4 + y2 − 6
5.
13. (x + 1)5
6. 7 x 3 + 56 y 3
14. (x − 1)6
7. 81r 4 − 16s 4
8. (x + y )2 + 2(x + y ) + 1
II. Simplification of Rational Algebraic Expressions
Simplify the following. Assume all variables are larger than zero.
1. 3 2 +5 − 4 + 4 0 4. 2 18 − 5 32 + 7 162
2. 9 ÷ 3 ⋅ 5 − 8 ÷ 2 + 27
81 6 x − 18 12x − 16
3. 5. ⋅
x4 3x + 2x − 8 4 x − 12
2




III. Solving Equations
A. Linear
1. 3 − 2(x − 1) = x − 10 3. y(y + 2 ) = y 2 − 6
2.
x x
− =1
4. 2[x − (1 − 3x )] = 3(x + 1)
2 7
B. Quadratic & Polynomial
 8  2 6. 3x 3 = 24
1.  y −  y +  = 0
7. (x + 1) + x = 25
2 2
 3  3
2. 2x − 4 x − 30 x = 0
3 2
8. 5y − y = 1
2

3. 27 x 3 = 1
4. (x − 3)(x + 6) = 9x + 22
5. t 2 + t +1 = 0




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, C. Rational
1 2 11 2 1
1. + =0 4. − =
y −1 y +1 x − 25 x − 5 x + 5
2


2 3 12 1 −6
2. − = 2 5. = 2
x−3 x +3 x −9 a a +5
1 2 5x −1 1 x
3. + = 2 6. = +
6 − x x + 3 x − 3x − 18 x − 3x x x − 3
2




D. Absolute value
1. 5 − 2z − 1 = 8 1 3 1
4. x− =
2. x + 5 − 7 = −2 2 4 4
5. y −1 = 7 + y
3. 5x − 1 = −2


E. Exponential
1. 10 x = 1000
3 x +5
4.
2
3x 9 x =( ) 1
2. 10 = 100 3
3. 2
x +1
=
1
5.
2
2 x 42x ( ) =
1
8
8

F. Logarithmic
1. log 2 (x + 5) = log 2 (1 − 5x ) 5. ln x + ln(x + 2 ) = ln 3
2. 2 log 3 (x + 1) = log 3 (4x ) 6. 3 = 4
2x x +1


3. log 2 (x + 1) + log 2 (x − 1) = 3
4. ln x + ln (2x + 1) = 0


G. Radicals
1. 4 2y − 1 − 2 = 0 4. x2 + 9 + x +1 = 0
2. 2x + 1 + 5 = 8 5. 3
3x + 2 + 4 = 6
3. 5x − 1 − 2 x + 1 = 0 6. 4
w2 + 7 = 2



IV. Solving Inequalities
Solve the following inequalities and express the answer graphically and using interval notation.
A. Linear
3 3. 3( x + 2 ) − 6 > −2( x − 3) + 14
1. x + 4 ≤ −2
5
2. 3( x + 3) ≥ 5( x − 1) 4. 2 ≤ 3 x − 10 ≤ 5


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, B. Absolute value: Solve and Graph.
1. 4x + 1 ≤ 6 x+5
3. ≥5
2. 4x + 3 + 2 > 9 3
4. 5 − 2 x < 15

C. Quadratic or Rational
1. 3 x 2 − 11x − 4 < 0 x+2
3. ≥0
2. 6x + 5x ≥ 4
2
3− x
4.
(x + 1)(x − 3) ≤ 0
2x + 7
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 + 3 y < −6 .

VI. Graphing Relations, Domain & Range
For each relation, state if it is a function, state the domain & range, and graph it.
y = x+2 6. x = y + 2
2
1.
y = x−2 7. y = x + 8 x − 6
2
2.
x −1
3. y= 8. y= −x
x+2
4. f (x ) = − x + 1 + 3 9. y = 3x
2x − 5 6x 2
5. f (x ) = 2 10. h (x ) =
x −9 3x 2 − 2 x − 1

VII. Exponents and Radicals
Simplify. Assume all variables are >0. Rationalize the denominators when needed.
−2
1.
3
− 8x 3  54a −6 b 2 
6.  −3 8

2. 5 147 − 4 48  9a b 
3. 5 15 − 3 ( ) 3
27a 3
3 7.
 x 3 y−3 
2 4
3
2a 2 b 2
4.  
 x − 53  2
  8.
5− 3
40x 4 x
5. 3
y9 9.
x +3




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