ACT Math: Algebra (Drill 4)

Question 1. What is the value of |−3x + 2| when x = 4?

• A) −10
• B) 10 ✓
• C) 14
• D) −14

Explanation: Substitute x = 4: −3(4) + 2 = −12 + 2 = −10. Then take the absolute value: |−10| = 10. Choice A results from computing the expression correctly but forgetting to apply the absolute value, leaving the answer as −10. Choice C results from computing |−3|(4) + 2 = 3(4) + 2 = 14, incorrectly applying the absolute value to only the coefficient before substituting. Choice D results from the same error as C but keeping the negative sign.

Question 2. Which of the following represents all values of x that satisfy −3x + 4 ≥ 13?

• A) x ≥ −3
• B) x ≤ −3 ✓
• C) x ≥ 3
• D) x ≤ 3

Explanation: Subtract 4 from both sides: −3x ≥ 9. Divide both sides by −3 and flip the inequality sign (required when dividing by a negative number): x ≤ −3. Choice A has the correct value but the wrong direction; it results from forgetting to flip the inequality when dividing by a negative. Choice C results from computing 13 − 4 = 9 and then dividing by 3 (positive) without accounting for the negative coefficient, giving x ≥ 3. Choice D has the correct direction of the inequality but uses the positive value 3 instead of −3.

Question 3. Which of the following is equivalent to (3x² − 5x + 2) − (x² + 3x − 6)?

• A) 2x² − 8x + 8 ✓
• B) 2x² − 2x − 4
• C) 2x² − 8x − 4
• D) 4x² − 2x − 4

Explanation: Distribute the subtraction sign across the second polynomial: (3x² − 5x + 2) − (x² + 3x − 6) = 3x² − 5x + 2 − x² − 3x + 6. Combine like terms: (3 − 1)x² + (−5 − 3)x + (2 + 6) = 2x² − 8x + 8. Choice B results from failing to distribute the negative to the −6 term: treating −(−6) as −6 instead of +6. Choice C results from distributing the negative correctly to the −6 but incorrectly keeping −4 rather than computing 2 + 6 = 8. Choice D results from adding the x² terms instead of subtracting: 3 + 1 = 4.

Question 4. If 9^{x − 1} = 27^{x + 1}, what is the value of x?

• A) −5 ✓
• B) −3
• C) 3
• D) 5

Explanation: Rewrite both sides using base 3: 9 = 3² and 27 = 3³. So 9^{x − 1} = (3²)^{x − 1} = 3^{2(x − 1)} = 3^{2x − 2}, and 27^{x + 1} = (3³)^{x + 1} = 3^{3(x + 1)} = 3^{3x + 3}. Set exponents equal: 2x − 2 = 3x + 3. Solve: −5 = x. Check: 9⁻⁶ = (3²)⁻⁶ = 3⁻¹² and 27⁻⁴ = (3³)⁻⁴ = 3⁻¹² ✓. Choice B results from a sign error when solving: adding 2 to both sides of 2x − 2 = 3x + 3 to get 2x = 3x + 5, then solving incorrectly. Choice C results from switching the sign of the answer. Choice D results from solving 2x − 2 = 3x + 3 as 2x − 3x = 3 + 2 → −x = 5, then forgetting the negative.

Question 5. Which of the following is equivalent to (x² + x − 6) / (x² − 4), for x ≠ 2 and x ≠ −2?

• A) (x + 3) / (x + 2) ✓
• B) (x − 3) / (x − 2)
• C) (x + 3) / (x − 2)
• D) (x − 3) / (x + 2)

Explanation: Factor the numerator: x² + x − 6 = (x + 3)(x − 2). Factor the denominator as a difference of squares: x² − 4 = (x + 2)(x − 2). The expression becomes (x + 3)(x − 2) / [(x + 2)(x − 2)]. Cancel the common factor (x − 2): the result is (x + 3) / (x + 2). Choice B results from incorrectly factoring the numerator as (x − 3)(x + 2), getting the signs wrong, and then canceling (x + 2). Choice C results from canceling (x + 2) instead of (x − 2) from the denominator. Choice D results from both factoring the numerator incorrectly and canceling the wrong factor.