MAT 101 Final Exam College Algebra Actual Exam 2026/2027
– Complete Exam-Style Questions with Detailed Rationales |
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Section A: Equations, Inequalities, & Linear Functions
Q1: A plumber charges a flat fee of $75 plus $45 per hour for repairs. If the total bill was
$345, which equation can be used to find the number of hours worked, and what is the
solution?
A. 45h = 345; h = 7.67 hours
B. 75 + 45h = 345; h = 6 hours [CORRECT]
C. 75h + 45 = 345; h = 4 hours
D. 45(h + 75) = 345; h = 1.67 hours
Correct Answer: B
Rationale: The total cost equals the flat fee plus the hourly rate times hours: 75 + 45h =
345. Solving: 45h = 270, so h = 6. Option A omits the flat fee, C incorrectly makes the fee
variable, and D incorrectly distributes the hourly rate.
Q2: Solve the absolute value inequality |2x − 5| ≤ 9. Which interval represents the
solution set?
A. (−∞, −2] ∪ [7, ∞)
B. [−2, 7] [CORRECT]
C. (−2, 7)
D. [2, 7]
Correct Answer: B
Rationale: |2x − 5| ≤ 9 becomes −9 ≤ 2x − 5 ≤ 9. Adding 5: −4 ≤ 2x ≤ 14. Dividing by 2: −2
≤ x ≤ 7. The "less than or equal to" symbol yields a closed interval [−2, 7]. Option A
represents the "greater than" case, C uses open endpoints incorrectly, and D has a sign
error.
,Q3: A line passes through the points (−3, 7) and (5, −1). What is the equation of the line
in slope-intercept form?
A. y = −x + 4 [CORRECT]
B. y = x + 4
C. y = −2x + 1
D. y = −x − 4
Correct Answer: A
Rationale: Slope m = (−1 − 7)/(5 − (−3)) = −8/8 = −1. Using point-slope with (5, −1): y −
(−1) = −1(x − 5), so y + 1 = −x + 5, thus y = −x + 4. Option B has the wrong sign for slope,
C has an incorrect slope, and D has an incorrect y-intercept.
Q4: Which equation represents a line perpendicular to 4x − 2y = 10 and passing through
the point (2, −3)?
A. y = −2x + 1
B. y = 2x − 7
C. y = −0.5x − 2 [CORRECT]
D. y = 0.5x − 4
Correct Answer: C
Rationale: Rewrite 4x − 2y = 10 as y = 2x − 5; slope is 2. A perpendicular line has slope
−1/2 (negative reciprocal). Using point-slope: y − (−3) = −0.5(x − 2), so y + 3 = −0.5x + 1,
thus y = −0.5x − 2. Option A has slope −2 (not negative reciprocal), B is parallel, and D
has the wrong intercept.
Q5: A company's profit function is P(x) = 12x − 4800, where x is the number of units
sold. How many units must be sold to break even?
A. 400 units [CORRECT]
B. 480 units
C. 4000 units
D. 57600 units
Correct Answer: A
, Rationale: Break-even occurs when P(x) = 0: 12x − 4800 = 0, so 12x = 4800, and x = 400.
Option B incorrectly divides 4800 by 10, C adds an extra zero, and D multiplies instead
of dividing.
Q6: Given f(x) = 3x² − 2x + 5 and g(x) = 4x − 1, find (f − g)(x).
A. 3x² + 2x + 6
B. 3x² − 6x + 6 [CORRECT]
C. 3x² + 2x + 4
D. 3x² − 6x + 4
Correct Answer: B
Rationale: (f − g)(x) = f(x) − g(x) = (3x² − 2x + 5) − (4x − 1) = 3x² − 2x + 5 − 4x + 1 = 3x² −
6x + 6. Option A incorrectly adds the linear terms, C has sign errors, and D omits the +1
from distributing the negative.
Q7: Find the inverse of the function f(x) = 2x − 7.
A. f⁻¹(x) = (x + 7)/2 [CORRECT]
B. f⁻¹(x) = 2x + 7
C. f⁻¹(x) = (x − 7)/2
D. f⁻¹(x) = x/2 + 7
Correct Answer: A
Rationale: To find the inverse, replace f(x) with y: y = 2x − 7. Swap x and y: x = 2y − 7.
Solve for y: x + 7 = 2y, so y = (x + 7)/2. Option B reverses operations incorrectly, C
subtracts instead of adds, and D misplaces the 7.
Q8: Evaluate the piecewise function f(x) = { x² + 1 if x < 2; 3x − 2 if x ≥ 2 } at x = 2.
A. 5
B. 4 [CORRECT]
C. 3
D. 2
Correct Answer: B
– Complete Exam-Style Questions with Detailed Rationales |
100% Verified | Pass Guaranteed – A+ Graded
Section A: Equations, Inequalities, & Linear Functions
Q1: A plumber charges a flat fee of $75 plus $45 per hour for repairs. If the total bill was
$345, which equation can be used to find the number of hours worked, and what is the
solution?
A. 45h = 345; h = 7.67 hours
B. 75 + 45h = 345; h = 6 hours [CORRECT]
C. 75h + 45 = 345; h = 4 hours
D. 45(h + 75) = 345; h = 1.67 hours
Correct Answer: B
Rationale: The total cost equals the flat fee plus the hourly rate times hours: 75 + 45h =
345. Solving: 45h = 270, so h = 6. Option A omits the flat fee, C incorrectly makes the fee
variable, and D incorrectly distributes the hourly rate.
Q2: Solve the absolute value inequality |2x − 5| ≤ 9. Which interval represents the
solution set?
A. (−∞, −2] ∪ [7, ∞)
B. [−2, 7] [CORRECT]
C. (−2, 7)
D. [2, 7]
Correct Answer: B
Rationale: |2x − 5| ≤ 9 becomes −9 ≤ 2x − 5 ≤ 9. Adding 5: −4 ≤ 2x ≤ 14. Dividing by 2: −2
≤ x ≤ 7. The "less than or equal to" symbol yields a closed interval [−2, 7]. Option A
represents the "greater than" case, C uses open endpoints incorrectly, and D has a sign
error.
,Q3: A line passes through the points (−3, 7) and (5, −1). What is the equation of the line
in slope-intercept form?
A. y = −x + 4 [CORRECT]
B. y = x + 4
C. y = −2x + 1
D. y = −x − 4
Correct Answer: A
Rationale: Slope m = (−1 − 7)/(5 − (−3)) = −8/8 = −1. Using point-slope with (5, −1): y −
(−1) = −1(x − 5), so y + 1 = −x + 5, thus y = −x + 4. Option B has the wrong sign for slope,
C has an incorrect slope, and D has an incorrect y-intercept.
Q4: Which equation represents a line perpendicular to 4x − 2y = 10 and passing through
the point (2, −3)?
A. y = −2x + 1
B. y = 2x − 7
C. y = −0.5x − 2 [CORRECT]
D. y = 0.5x − 4
Correct Answer: C
Rationale: Rewrite 4x − 2y = 10 as y = 2x − 5; slope is 2. A perpendicular line has slope
−1/2 (negative reciprocal). Using point-slope: y − (−3) = −0.5(x − 2), so y + 3 = −0.5x + 1,
thus y = −0.5x − 2. Option A has slope −2 (not negative reciprocal), B is parallel, and D
has the wrong intercept.
Q5: A company's profit function is P(x) = 12x − 4800, where x is the number of units
sold. How many units must be sold to break even?
A. 400 units [CORRECT]
B. 480 units
C. 4000 units
D. 57600 units
Correct Answer: A
, Rationale: Break-even occurs when P(x) = 0: 12x − 4800 = 0, so 12x = 4800, and x = 400.
Option B incorrectly divides 4800 by 10, C adds an extra zero, and D multiplies instead
of dividing.
Q6: Given f(x) = 3x² − 2x + 5 and g(x) = 4x − 1, find (f − g)(x).
A. 3x² + 2x + 6
B. 3x² − 6x + 6 [CORRECT]
C. 3x² + 2x + 4
D. 3x² − 6x + 4
Correct Answer: B
Rationale: (f − g)(x) = f(x) − g(x) = (3x² − 2x + 5) − (4x − 1) = 3x² − 2x + 5 − 4x + 1 = 3x² −
6x + 6. Option A incorrectly adds the linear terms, C has sign errors, and D omits the +1
from distributing the negative.
Q7: Find the inverse of the function f(x) = 2x − 7.
A. f⁻¹(x) = (x + 7)/2 [CORRECT]
B. f⁻¹(x) = 2x + 7
C. f⁻¹(x) = (x − 7)/2
D. f⁻¹(x) = x/2 + 7
Correct Answer: A
Rationale: To find the inverse, replace f(x) with y: y = 2x − 7. Swap x and y: x = 2y − 7.
Solve for y: x + 7 = 2y, so y = (x + 7)/2. Option B reverses operations incorrectly, C
subtracts instead of adds, and D misplaces the 7.
Q8: Evaluate the piecewise function f(x) = { x² + 1 if x < 2; 3x − 2 if x ≥ 2 } at x = 2.
A. 5
B. 4 [CORRECT]
C. 3
D. 2
Correct Answer: B
