CLST 103 - Liberty University
Mathematics Assessment — Part 2
Comprehensive Competency Examination
65 Questions • 9 Sections • Multiple Choice with Step-by-Step Rationales
Instructions: This assessment evaluates mathematical competency across nine core topic areas aligned with the
CLST 103 curriculum. Each question presents four answer choices (A–D) with exactly one correct response. The
correct answer is marked [CORRECT] and is followed by a complete step-by-step rationale showing the
procedure used and explaining the specific mathematical error in each distractor. Distractors are designed to
identify common student misconceptions including sign errors, order-of-operations mistakes, incorrect factoring,
misapplied formulas, and extraneous solutions. Recommended time: 90 minutes.
Section 1: Linear Equations, Inequalities, and Absolute Value
Q1. A car rental company charges a flat fee of $35 plus $0.25 per mile driven. If a customer's total bill was
$92.50, how many miles did the customer drive?
A. 230 miles [CORRECT]
B. 250 miles
C. 270 miles
D. 230.5 miles
Correct Answer: A
Rationale: Let m = miles driven. The cost equation is 35 + 0.25m = 92.50. Subtract 35 from both sides: 0.25m =
57.50. Divide by 0.25: m = 57..25 = 230. The customer drove 230 miles. Choice B (250) results from dividing
57.50 by 0.23 instead of 0.25 (using the wrong divisor). Choice C (270) results from adding the flat fee instead of
subtracting (92.50 + 35 = 127.50, then / 0.25 ≈ 510 — partial error). Choice D (230.5) results from an arithmetic
rounding mistake during division.
Q2. Solve the literal equation 2x + 3y = 12 for y in terms of x.
A. y = (12 - 2x)/3
B. y = (12 - 2x)/(-3)
C. y = (2x - 12)/3
D. y = 4 - (2/3)x [CORRECT]
Correct Answer: D
Rationale: Starting with 2x + 3y = 12, isolate the y-term: subtract 2x from both sides to obtain 3y = 12 - 2x. Divide
every term by 3: y = (12 - 2x)/3 = 4 - (2/3)x. Choice A is the unsimplified form (technically correct but not fully
reduced); however, Choice D is the most simplified equivalent form, so D is the best answer. Choice B incorrectly
changes the sign of the denominator. Choice C incorrectly writes the numerator as (2x - 12), which equals -(12 - 2x),
giving the wrong sign on every term.
Q3. Solve the inequality -3(x - 4) ≤ 15 and express the solution in interval notation.
Liberty University - CLST 103 Mathematics Competency Assessment
,CLST 103 - Math Assessment Part 2 Page 2
A. (-∞, -1]
B. [-1, ∞) [CORRECT]
C. (-∞, 9]
D. [9, ∞)
Correct Answer: B
Rationale: Distribute the -3: -3x + 12 ≤ 15. Subtract 12 from both sides: -3x ≤ 3. Divide both sides by -3, which
reverses the inequality: x ≥ -1. In interval notation this is [-1, ∞). Choice A results from forgetting to reverse the
inequality sign when dividing by a negative number. Choice C (x ≤ 9) results from incorrectly distributing -3 as -3x -
12 (sign error in distribution). Choice D results from compounding multiple sign errors.
Q4. Solve the compound inequality -5 ≤ 2x + 1 < 9 and express the solution in interval notation.
A. (-∞, 3)
B. [-3, 4) [CORRECT]
C. (-3, 4]
D. [-3, 4]
Correct Answer: B
Rationale: Split into two parts and solve simultaneously. Subtract 1 from each section: -6 ≤ 2x < 8. Divide each part
by 2: -3 ≤ x < 4. In interval notation this is [-3, 4). Choice A incorrectly treats the lower bound as open and unbounded
below. Choice C reverses which endpoint is open vs. closed (a common error when reading the original inequality
direction). Choice D uses closed brackets on both ends, forgetting that the original right-end (<) is strict and therefore
open.
Q5. Solve the compound inequality x < -2 OR x ≥ 3. Which number line correctly represents the solution?
A. All real numbers
B. (-2, 3]
C. (-∞, -2) ∪ [3, ∞) [CORRECT]
D. [-2, 3)
Correct Answer: C
Rationale: The inequality uses OR, so the solution is the union of the two intervals: (-∞, -2) ∪ [3, ∞). The parenthesis
at -2 indicates x < -2 (open), and the bracket at 3 indicates x ≥ 3 (closed). Choice A incorrectly assumes the union
covers all reals. Choice B incorrectly uses the AND interpretation (the intersection). Choice D reverses both the
inequality direction and which endpoint is open/closed.
Q6. Solve the absolute value equation |2x - 5| = 11.
A. x = 3 or x = -3
B. x = 8 or x = -3 [CORRECT]
C. x = 8 or x = 3
D. x = -8 or x = 3
Correct Answer: B
Rationale: Set up two equations: 2x - 5 = 11 and 2x - 5 = -11. Solving the first: 2x = 16, x = 8. Solving the second: 2x
= -6, x = -3. The solution set is {8, -3}. Choice A results from incorrectly splitting |2x - 5| = 11 as 2x - 5 = ±3 instead
of ±11 (numerical error). Choice C results from mistakenly solving the second equation as 2x - 5 = 1 (dropping the
Liberty University - CLST 103 Mathematics Competency Assessment
, CLST 103 - Math Assessment Part 2 Page 3
negative). Choice D results from a sign error in solving the first equation: 2x - 5 = 11 mis-solved as 2x = -16, x = -8.
Q7. Solve the absolute value inequality |x + 4| ≥ 7 and express the solution in interval notation.
A. (-∞, -11] ∪ [3, ∞) [CORRECT]
B. [-11, 3]
C. (-∞, -3] ∪ [11, ∞)
D. (-3, 11)
Correct Answer: A
Rationale: Because |x + 4| ≥ 7 means the quantity is 7 or more from zero, set up x + 4 ≤ -7 OR x + 4 ≥ 7. Solve each:
x ≤ -11 OR x ≥ 3. The solution in interval notation is (-∞, -11] ∪ [3, ∞). Choice B incorrectly applies the AND
interpretation (used for |expression| ≤ value). Choice C reverses the constants -11 and 3 (a common swap error when
applying the negative branch). Choice D mixes up the inequality direction and the boundary constants.
Q8. A manufacturing process produces rods that must be 24 cm long with a tolerance of 0.05 cm. Write the
absolute value inequality that represents the acceptable range of lengths L for the rods.
A. |L - 24| ≤ 0.05 [CORRECT]
B. |L + 24| ≤ 0.05
C. |L - 24| ≥ 0.05
D. |L - 0.05| ≤ 24
Correct Answer: A
Rationale: The target length is 24 cm and the tolerance (maximum acceptable deviation) is 0.05 cm. The acceptable
lengths are within 0.05 cm of 24, so |L - 24| ≤ 0.05. Choice B uses (L + 24) instead of (L - 24), which measures
distance from -24, not 24. Choice C reverses the inequality, which would require rods to be at least 0.05 cm away
from 24 (the opposite of acceptable tolerance). Choice D confuses the role of the constants, treating 0.05 as the center
and 24 as the tolerance.
Section 2: Graphing Linear Equations and Functions
Q9. Find the slope of the line passing through the points (-3, 7) and (2, -8).
A. -3 [CORRECT]
B. 3
C. -1/3
D. 1/3
Correct Answer: A
Rationale: Using the slope formula m = (y2 - y1)/(x2 - x1) = (-8 - 7)/(2 - (-3)) = -15/5 = -3. The slope is -3. Choice B
(positive 3) results from reversing the order of subtraction in only the numerator or denominator but not both,
producing the wrong sign. Choice C (-1/3) results from inverting the fraction -15/5 (a common arithmetic-swap error).
Choice D (1/3) compounds both the inversion and the sign error.
Q10. Convert the equation 3x - 4y = 12 into slope-intercept form (y = mx + b).
A. y = (3/4)x - 3 [CORRECT]
B. y = -(3/4)x + 3
Liberty University - CLST 103 Mathematics Competency Assessment
Mathematics Assessment — Part 2
Comprehensive Competency Examination
65 Questions • 9 Sections • Multiple Choice with Step-by-Step Rationales
Instructions: This assessment evaluates mathematical competency across nine core topic areas aligned with the
CLST 103 curriculum. Each question presents four answer choices (A–D) with exactly one correct response. The
correct answer is marked [CORRECT] and is followed by a complete step-by-step rationale showing the
procedure used and explaining the specific mathematical error in each distractor. Distractors are designed to
identify common student misconceptions including sign errors, order-of-operations mistakes, incorrect factoring,
misapplied formulas, and extraneous solutions. Recommended time: 90 minutes.
Section 1: Linear Equations, Inequalities, and Absolute Value
Q1. A car rental company charges a flat fee of $35 plus $0.25 per mile driven. If a customer's total bill was
$92.50, how many miles did the customer drive?
A. 230 miles [CORRECT]
B. 250 miles
C. 270 miles
D. 230.5 miles
Correct Answer: A
Rationale: Let m = miles driven. The cost equation is 35 + 0.25m = 92.50. Subtract 35 from both sides: 0.25m =
57.50. Divide by 0.25: m = 57..25 = 230. The customer drove 230 miles. Choice B (250) results from dividing
57.50 by 0.23 instead of 0.25 (using the wrong divisor). Choice C (270) results from adding the flat fee instead of
subtracting (92.50 + 35 = 127.50, then / 0.25 ≈ 510 — partial error). Choice D (230.5) results from an arithmetic
rounding mistake during division.
Q2. Solve the literal equation 2x + 3y = 12 for y in terms of x.
A. y = (12 - 2x)/3
B. y = (12 - 2x)/(-3)
C. y = (2x - 12)/3
D. y = 4 - (2/3)x [CORRECT]
Correct Answer: D
Rationale: Starting with 2x + 3y = 12, isolate the y-term: subtract 2x from both sides to obtain 3y = 12 - 2x. Divide
every term by 3: y = (12 - 2x)/3 = 4 - (2/3)x. Choice A is the unsimplified form (technically correct but not fully
reduced); however, Choice D is the most simplified equivalent form, so D is the best answer. Choice B incorrectly
changes the sign of the denominator. Choice C incorrectly writes the numerator as (2x - 12), which equals -(12 - 2x),
giving the wrong sign on every term.
Q3. Solve the inequality -3(x - 4) ≤ 15 and express the solution in interval notation.
Liberty University - CLST 103 Mathematics Competency Assessment
,CLST 103 - Math Assessment Part 2 Page 2
A. (-∞, -1]
B. [-1, ∞) [CORRECT]
C. (-∞, 9]
D. [9, ∞)
Correct Answer: B
Rationale: Distribute the -3: -3x + 12 ≤ 15. Subtract 12 from both sides: -3x ≤ 3. Divide both sides by -3, which
reverses the inequality: x ≥ -1. In interval notation this is [-1, ∞). Choice A results from forgetting to reverse the
inequality sign when dividing by a negative number. Choice C (x ≤ 9) results from incorrectly distributing -3 as -3x -
12 (sign error in distribution). Choice D results from compounding multiple sign errors.
Q4. Solve the compound inequality -5 ≤ 2x + 1 < 9 and express the solution in interval notation.
A. (-∞, 3)
B. [-3, 4) [CORRECT]
C. (-3, 4]
D. [-3, 4]
Correct Answer: B
Rationale: Split into two parts and solve simultaneously. Subtract 1 from each section: -6 ≤ 2x < 8. Divide each part
by 2: -3 ≤ x < 4. In interval notation this is [-3, 4). Choice A incorrectly treats the lower bound as open and unbounded
below. Choice C reverses which endpoint is open vs. closed (a common error when reading the original inequality
direction). Choice D uses closed brackets on both ends, forgetting that the original right-end (<) is strict and therefore
open.
Q5. Solve the compound inequality x < -2 OR x ≥ 3. Which number line correctly represents the solution?
A. All real numbers
B. (-2, 3]
C. (-∞, -2) ∪ [3, ∞) [CORRECT]
D. [-2, 3)
Correct Answer: C
Rationale: The inequality uses OR, so the solution is the union of the two intervals: (-∞, -2) ∪ [3, ∞). The parenthesis
at -2 indicates x < -2 (open), and the bracket at 3 indicates x ≥ 3 (closed). Choice A incorrectly assumes the union
covers all reals. Choice B incorrectly uses the AND interpretation (the intersection). Choice D reverses both the
inequality direction and which endpoint is open/closed.
Q6. Solve the absolute value equation |2x - 5| = 11.
A. x = 3 or x = -3
B. x = 8 or x = -3 [CORRECT]
C. x = 8 or x = 3
D. x = -8 or x = 3
Correct Answer: B
Rationale: Set up two equations: 2x - 5 = 11 and 2x - 5 = -11. Solving the first: 2x = 16, x = 8. Solving the second: 2x
= -6, x = -3. The solution set is {8, -3}. Choice A results from incorrectly splitting |2x - 5| = 11 as 2x - 5 = ±3 instead
of ±11 (numerical error). Choice C results from mistakenly solving the second equation as 2x - 5 = 1 (dropping the
Liberty University - CLST 103 Mathematics Competency Assessment
, CLST 103 - Math Assessment Part 2 Page 3
negative). Choice D results from a sign error in solving the first equation: 2x - 5 = 11 mis-solved as 2x = -16, x = -8.
Q7. Solve the absolute value inequality |x + 4| ≥ 7 and express the solution in interval notation.
A. (-∞, -11] ∪ [3, ∞) [CORRECT]
B. [-11, 3]
C. (-∞, -3] ∪ [11, ∞)
D. (-3, 11)
Correct Answer: A
Rationale: Because |x + 4| ≥ 7 means the quantity is 7 or more from zero, set up x + 4 ≤ -7 OR x + 4 ≥ 7. Solve each:
x ≤ -11 OR x ≥ 3. The solution in interval notation is (-∞, -11] ∪ [3, ∞). Choice B incorrectly applies the AND
interpretation (used for |expression| ≤ value). Choice C reverses the constants -11 and 3 (a common swap error when
applying the negative branch). Choice D mixes up the inequality direction and the boundary constants.
Q8. A manufacturing process produces rods that must be 24 cm long with a tolerance of 0.05 cm. Write the
absolute value inequality that represents the acceptable range of lengths L for the rods.
A. |L - 24| ≤ 0.05 [CORRECT]
B. |L + 24| ≤ 0.05
C. |L - 24| ≥ 0.05
D. |L - 0.05| ≤ 24
Correct Answer: A
Rationale: The target length is 24 cm and the tolerance (maximum acceptable deviation) is 0.05 cm. The acceptable
lengths are within 0.05 cm of 24, so |L - 24| ≤ 0.05. Choice B uses (L + 24) instead of (L - 24), which measures
distance from -24, not 24. Choice C reverses the inequality, which would require rods to be at least 0.05 cm away
from 24 (the opposite of acceptable tolerance). Choice D confuses the role of the constants, treating 0.05 as the center
and 24 as the tolerance.
Section 2: Graphing Linear Equations and Functions
Q9. Find the slope of the line passing through the points (-3, 7) and (2, -8).
A. -3 [CORRECT]
B. 3
C. -1/3
D. 1/3
Correct Answer: A
Rationale: Using the slope formula m = (y2 - y1)/(x2 - x1) = (-8 - 7)/(2 - (-3)) = -15/5 = -3. The slope is -3. Choice B
(positive 3) results from reversing the order of subtraction in only the numerator or denominator but not both,
producing the wrong sign. Choice C (-1/3) results from inverting the fraction -15/5 (a common arithmetic-swap error).
Choice D (1/3) compounds both the inversion and the sign error.
Q10. Convert the equation 3x - 4y = 12 into slope-intercept form (y = mx + b).
A. y = (3/4)x - 3 [CORRECT]
B. y = -(3/4)x + 3
Liberty University - CLST 103 Mathematics Competency Assessment
