CLEP College Algebra — Questions and Answers

Question 1: Solve: 4x + 1 ≥ 2x + 9

x ≥ 4 (Correct answer)
x ≥ 2
x ≤ 4
x ≤ 2

Correct answer: x ≥ 4

Subtract 2x to get 2x + 1 ≥ 9, subtract 1 to get 2x ≥ 8, divide to get x ≥ 4.

Question 2: Which expression is equivalent to x^(3/2)?

x²√x
√(x²)
√x
x√x (Correct answer)

Correct answer: x√x

x^(3/2) = x^1 · x^(1/2) = x · √x = x√x.

Question 3: What is the value of 3⁻²?

-9
1/6
-1/9
1/9 (Correct answer)

Correct answer: 1/9

A negative exponent means reciprocal: 3⁻² = 1/3² = 1/9.

Question 4: Solve x² - 9 = 0 by factoring.

x = ±3 (Correct answer)
x = ±9
x = -3 only
x = 3 only

Correct answer: x = ±3

Factor as (x - 3)(x + 3) = 0, giving x = 3 or x = -3.

Question 5: A system with exactly one solution is called:

Consistent and independent (Correct answer)
Dependent
Inconsistent
Trivial

Correct answer: Consistent and independent

A system with exactly one solution is consistent (a solution exists) and independent (distinct lines).

Question 6: What transformation does y = f(x - 2) represent?

Shift 2 units right (Correct answer)
Shift 2 units up
Shift 2 units left
Shift 2 units down

Correct answer: Shift 2 units right

Replacing x with (x - 2) shifts the graph 2 units to the right.

Question 7: An even function satisfies:

f(0) = 0
f(x + 1) = f(x)
f(-x) = -f(x)
f(-x) = f(x) (Correct answer)

Correct answer: f(-x) = f(x)

An even function satisfies f(-x) = f(x), meaning it is symmetric about the y-axis.

Question 8: If f(x) = 5 for all x, the function is called a:

Step function
Quadratic function
Constant function (Correct answer)
Linear function

Correct answer: Constant function

A function that outputs the same value for every input is a constant function.

Question 9: Simplify: √(x⁴), assuming x ≥ 0

x³
x
x² (Correct answer)
x⁴

Correct answer: x²

√(x⁴) = x^(4/2) = x².

Question 10: Solve x² + 2x - 15 = 0 by factoring.

x = 3 and x = 5
x = 3 and x = -5 (Correct answer)
x = -3 and x = 5
x = 5 and x = -3

Correct answer: x = 3 and x = -5

Factor as (x + 5)(x - 3) = 0, giving x = 3 and x = -5.

Question 11: Subtract: (5x² - 3x + 2) - (2x² + x - 4)

3x² - 4x + 6 (Correct answer)
7x² - 2x - 2
3x² + 4x - 6
3x² - 4x - 2

Correct answer: 3x² - 4x + 6

Distribute the negative: (5-2)x² = 3x², (-3-1)x = -4x, (2+4) = 6, giving 3x² - 4x + 6.

Question 12: Solve: 2x - 3 < 7

x < 2
x < 5 (Correct answer)
x > 5
x > 2

Correct answer: x < 5

Add 3 to both sides to get 2x < 10, then divide by 2 to get x < 5.

Question 13: Simplify: (3x² + 2x - 1) + (x² - 4x + 5)

2x² - 2x + 4
4x² + 6x - 6
4x² - 2x + 4 (Correct answer)
4x² - 2x - 4

Correct answer: 4x² - 2x + 4

Combine like terms: (3+1)x² = 4x², (2-4)x = -2x, (-1+5) = 4, giving 4x² - 2x + 4.

Question 14: When must you flip the inequality symbol while solving?

When subtracting a positive number
When adding a positive number
When adding a negative number
When multiplying or dividing by a negative number (Correct answer)

Correct answer: When multiplying or dividing by a negative number

Multiplying or dividing both sides by a negative number reverses the inequality direction.

Question 15: What are the solutions to x² - 5x + 6 = 0?

x = 1 and x = 6
x = 2 and x = 3 (Correct answer)
x = -2 and x = -3
x = -1 and x = -6

Correct answer: x = 2 and x = 3

Factor as (x - 2)(x - 3) = 0, giving x = 2 and x = 3.

Question 16: Simplify: (x²)³

2x⁶
x⁸
x⁵
x⁶ (Correct answer)

Correct answer: x⁶

When raising a power to a power, multiply the exponents: (x²)³ = x^(2×3) = x⁶.

Question 17: Which of the following relations is a function?

{(0,0),(1,1),(1,-1)}
{(1,2),(2,3),(3,4)} (Correct answer)
{(1,2),(1,3),(2,4)}
{(2,1),(2,3),(2,5)}

Correct answer: {(1,2),(2,3),(3,4)}

A function requires each input to have exactly one output; only the second set has no repeated x-values.

Question 18: Solve: 2x + 3y = 12 and x = 3

y = 2 (Correct answer)
y = 3
y = 4
y = 1

Correct answer: y = 2

Substitute x = 3: 6 + 3y = 12, so 3y = 6 and y = 2.

Question 19: If f(x) = 3x - 2, what is f(4)?

10 (Correct answer)
12
6
8

Correct answer: 10

f(4) = 3(4) - 2 = 10.

Question 20: A negative leading coefficient on a quadratic function:

Makes it narrower
Shifts it right
Makes it open downward (Correct answer)
Shifts the parabola up

Correct answer: Makes it open downward

A negative leading coefficient reflects the parabola over the x-axis so it opens downward.

Question 21: What is the inverse of f(x) = 2x + 6?

f⁻¹(x) = x/2 + 6
f⁻¹(x) = (x + 6)/2
f⁻¹(x) = 2x - 6
f⁻¹(x) = (x - 6)/2 (Correct answer)

Correct answer: f⁻¹(x) = (x - 6)/2

Swap x and y: x = 2y + 6, then solve for y: y = (x - 6)/2.

Question 22: What notation means 'f composed with g'?

f ÷ g
f ∘ g (Correct answer)
f + g
f · g

Correct answer: f ∘ g

Function composition is written f ∘ g, meaning f(g(x)).

Question 23: Expand: (x + 4)²

x² + 8x + 16 (Correct answer)
x² + 16
x² + 4x + 16
x² - 8x + 16

Correct answer: x² + 8x + 16

(x + 4)² = x² + 2(4)x + 4² = x² + 8x + 16.

Question 24: If f(x) = x² + 1 and g(x) = 2x, what is (f ∘ g)(3)?

13
37 (Correct answer)
7
19

Correct answer: 37

g(3) = 6; then f(6) = 36 + 1 = 37.

Question 25: Which describes an increasing function on an interval?

As x increases, y increases (Correct answer)
As x decreases, y increases
As x increases, y decreases
As x increases, y stays constant

Correct answer: As x increases, y increases

An increasing function produces larger outputs as input values increase.

Question 26: How many terms does (a + b)(c + d) produce before combining like terms?

3
4 (Correct answer)
6
2

Correct answer: 4

Using FOIL gives exactly 4 terms: ac, ad, bc, and bd.

Question 27: If y = 2x + 1, what is y when x = 4?

10
7
9 (Correct answer)
8

Correct answer: 9

Substitute x = 4 to get y = 2(4) + 1 = 9.

Question 28: Factor: 2x² + 8x

2(x² + 8x)
4x(x + 2)
2x(x + 4) (Correct answer)
x(2x + 8)

Correct answer: 2x(x + 4)

The GCF is 2x, so 2x² + 8x = 2x(x + 4).

Question 29: What is the vertex form of a quadratic equation?

y = (ax + b)(cx + d)
y = mx + b
y = a(x - h)² + k (Correct answer)
y = ax² + bx + c

Correct answer: y = a(x - h)² + k

Vertex form is y = a(x - h)² + k, where (h, k) is the vertex of the parabola.

Question 30: What is (2x²)(3x³)?

6x⁵ (Correct answer)
5x⁵
5x⁶
6x⁶

Correct answer: 6x⁵

Multiply coefficients 2 × 3 = 6 and add exponents 2 + 3 = 5, giving 6x⁵.

Question 31: Which equation is parallel to y = 5x + 2?

y = 2x + 5
y = -5x + 2
y = 5x - 7 (Correct answer)
y = (1/5)x + 2

Correct answer: y = 5x - 7

Parallel lines have the same slope; y = 5x - 7 also has slope 5.

Question 32: The vertical line test determines whether a graph is:

A linear equation
A function (Correct answer)
A polynomial
An inequality

Correct answer: A function

The vertical line test checks if any vertical line intersects the graph more than once; if not, it is a function.

Question 33: Identify the constant term in: 7x⁴ - 3x² + 9

7
9 (Correct answer)
-3
4

Correct answer: 9

The constant term is the term with no variable, which is 9.

Question 34: Simplify: (x³ · x⁻¹) / x²

1 (Correct answer)
x⁻²
x⁴
x²

Correct answer: 1

x³ · x⁻¹ = x², then x² / x² = x⁰ = 1.

Question 35: What is the equation of a line with slope 3 passing through (0, -1)?

y = x - 3
y = 3x - 1 (Correct answer)
y = 3x + 1
y = -x + 3

Correct answer: y = 3x - 1

Using y = mx + b with m = 3 and b = -1 gives y = 3x - 1.

Question 36: Factor 4x² + 12x + 9.

(4x + 3)(x + 3)
(4x + 9)(x + 1)
(2x + 3)² (Correct answer)
(2x + 9)(2x + 1)

Correct answer: (2x + 3)²

This is a perfect square trinomial: (2x)² + 2(2x)(3) + 3² = (2x + 3)².

Question 37: What is the domain of f(x) = 1/(x - 5)?

All real numbers
x > 5
x ≠ 5 (Correct answer)
x ≠ 0

Correct answer: x ≠ 5

The denominator cannot be zero: x - 5 ≠ 0, so x ≠ 5.

Question 38: What is the degree of 4x³ - 2x² + 7x - 1?

3 (Correct answer)
2
4
1

Correct answer: 3

The degree is the highest exponent, which is 3.

Question 39: Factor out the GCF: 6x³ + 9x².

3x²(2x + 3) (Correct answer)
3x(2x² + 3x)
6x²(x + 9)
9x²(x + 1)

Correct answer: 3x²(2x + 3)

The GCF of 6x³ and 9x² is 3x², giving 3x²(2x + 3).

Question 40: What is the domain of f(x) = √(x - 3)?

x ≥ 0
x > 3
All real numbers
x ≥ 3 (Correct answer)

Correct answer: x ≥ 3

The radicand must be non-negative: x - 3 ≥ 0, so x ≥ 3.

Question 41: What is 27^(1/3)?

9
3 (Correct answer)
81
6

Correct answer: 3

27^(1/3) is the cube root of 27; since 3³ = 27, the answer is 3.

Question 42: Factor: x² - 2x - 8.

(x - 4)(x - 2)
(x + 4)(x + 2)
(x + 4)(x - 2)
(x - 4)(x + 2) (Correct answer)

Correct answer: (x - 4)(x + 2)

Find two numbers that multiply to -8 and add to -2: -4 and 2, giving (x - 4)(x + 2).

Question 43: Which function graphs as a straight line?

f(x) = x²
f(x) = 2x + 1 (Correct answer)
f(x) = 1/x
f(x) = √x

Correct answer: f(x) = 2x + 1

f(x) = 2x + 1 is a linear function and graphs as a straight line.

Question 44: Solve: |x - 2| > 3

x < -1 only
-1 < x < 5
x > 5 or x < -1 (Correct answer)
x > 5 only

Correct answer: x > 5 or x < -1

|x - 2| > 3 means x - 2 > 3 or x - 2 < -3, giving x > 5 or x < -1.

Question 45: What is the result of (2x + 3)(3x - 1)?

6x² - 7x + 3
6x² - 7x - 3
6x² + 7x - 3 (Correct answer)
6x² + 7x + 3

Correct answer: 6x² + 7x - 3

FOIL: 6x² - 2x + 9x - 3 = 6x² + 7x - 3.

Question 46: Multiply: (x + 3)(x - 2).

x² + x - 6 (Correct answer)
x² + 5x - 6
x² - 5x + 6
x² - x - 6

Correct answer: x² + x - 6

Use FOIL: x² - 2x + 3x - 6 = x² + x - 6.

Question 47: Solve the compound inequality: -1 < x + 2 < 5

1 < x < 3
-3 < x < 3 (Correct answer)
-3 < x < 7
1 < x < 7

Correct answer: -3 < x < 3

Subtract 2 from all three parts: -3 < x < 3.

Question 48: Solve: 3x ≤ 12

x ≥ 4
x ≤ 4 (Correct answer)
x ≥ 9
x ≤ 9

Correct answer: x ≤ 4

Divide both sides by 3 to get x ≤ 4.

Question 49: When a system of two linear equations has no solution, geometrically the lines are:

Intersecting at one point
The same line
Parallel (Correct answer)
Perpendicular

Correct answer: Parallel

No solution means the lines are parallel and never intersect.

Question 50: What is the range of f(x) = x² for all real x?

y ≥ 0 (Correct answer)
y > 0
y ≤ 0
All real numbers

Correct answer: y ≥ 0

Since x² ≥ 0 for all real x, the range is y ≥ 0.

Question 51: Which expression is equivalent to ∛(x⁶)?

x³
x² (Correct answer)
x⁹
x¹⁸

Correct answer: x²

∛(x⁶) = x^(6/3) = x².

Question 52: What is the product √3 · √12?

9
3
12
6 (Correct answer)

Correct answer: 6

√3 · √12 = √(3 × 12) = √36 = 6.

Question 53: What transformation does y = f(x) + 3 represent?

Shift 3 units left
Shift 3 units down
Shift 3 units right
Shift 3 units up (Correct answer)

Correct answer: Shift 3 units up

Adding a constant outside the function shifts the graph vertically upward by 3 units.

Question 54: Multiply: (x - 4)(x + 4).

x² - 16 (Correct answer)
x² + 8x - 16
x² + 16
x² - 8x + 16

Correct answer: x² - 16

This is a difference of squares: (a - b)(a + b) = a² - b² = x² - 16.

Question 55: Which form of a quadratic is easiest for reading the roots directly?

Point-slope form
Factored form (Correct answer)
Standard form
Vertex form

Correct answer: Factored form

Factored form y = a(x - r₁)(x - r₂) directly shows the roots r₁ and r₂.

Question 56: What is the value of 8^(2/3)?

2
4 (Correct answer)
16
8

Correct answer: 4

8^(2/3) = (8^(1/3))² = 2² = 4.

Question 57: Simplify: x³ · x⁴

x¹²
2x⁷
x⁷ (Correct answer)
x

Correct answer: x⁷

When multiplying powers with the same base, add the exponents: x³ · x⁴ = x^(3+4) = x⁷.

Question 58: What is the x-intercept of f(x) = 2x - 8?

x = 4 (Correct answer)
x = 2
x = -8
x = -4

Correct answer: x = 4

Set f(x) = 0: 2x - 8 = 0 gives x = 4.

Question 59: Simplify: (2x²)³

8x⁵
2x⁵
8x⁶ (Correct answer)
6x⁶

Correct answer: 8x⁶

Apply the power to each factor: 2³ = 8 and (x²)³ = x⁶, giving 8x⁶.

Question 60: Rationalize the denominator of 1/√5.

1/5
√5/25
√5/5 (Correct answer)
5/√5

Correct answer: √5/5

Multiply by √5/√5: (1 · √5)/(√5 · √5) = √5/5.

Question 61: Solve: 5 - 2x < 1

x < -2
x < 2
x > -2
x > 2 (Correct answer)

Correct answer: x > 2

Subtract 5: -2x < -4, then divide by -2 and flip: x > 2.

CLEP College Algebra

The CLEP College Algebra exam tests mastery of algebraic concepts typically covered in a one-semester college algebra course, including functions, equations, inequalities, and algebraic operations. A passing score can earn undergraduate college credit at thousands of institutions.

Questions

90 min

Time Limit

70%

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