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Advanced Algebra Practice Exam
Question 1: Solve for x: 2x + 3 = 11. A. x = 4 B. x = 3 C. x = 5 D. x = 6 Answer: A Explanation: Subtract 3 from both sides to obtain 2x = 8; dividing by 2 gives x = 4. Question 2: Solve the inequality: 3x – 5 > 7. A. x > 4 B. x > 5 C. x > 3 D. x > 6 Answer: A Explanation: Add 5 to both sides to get 3x > 12; then divide by 3 to find x > 4. Question 3: Solve for x: 5(x – 2) = 3x + 4. A. x = 5 B. x = 6 C. x = 7 D. x = 8 Answer: C Explanation: Distribute to obtain 5x – 10 = 3x + 4; subtract 3x to get 2x – 10 = 4, then add 10 and divide by 2 to yield x = 7. Question 4: Solve for x: 4 – 2x = 10. A. x = – 3 B. x = 3 C. x = – 2 D. x = 2 Answer: A Explanation: Subtract 4 to obtain – 2x = 6; dividing by – 2 gives x = – 3. Question 5: Solve the equation: (7x)/3 = 14. A. x = 4 B. x = 5
C. x = 6 D. x = 7 Answer: C Explanation: Multiply both sides by 3 to get 7x = 42; dividing by 7 results in x = 6. Question 6: Solve for x: – 3x + 9 = 0. A. x = – 3 B. x = 3 C. x = 0 D. x = 9 Answer: B Explanation: Subtract 9 to obtain – 3x = – 9; dividing by – 3 gives x = 3. Question 7: Solve the inequality: – 2x + 4 ≤ 8. A. x ≤ – 2 B. x ≥ – 2 C. x < – 2 D. x > – 2 Answer: B Explanation: Subtract 4 to get – 2x ≤ 4; dividing by – 2 (and reversing the inequality) yields x ≥ –
Question 8: Solve for x: 6x + 2 = 2(2x + 5). A. x = 2 B. x = 3 C. x = 4 D. x = 5 Answer: C Explanation: Expand the right side to obtain 6x + 2 = 4x + 10; subtract 4x to get 2x + 2 = 10, then subtract 2 and divide by 2 to find x = 4. Question 9: Find x if 4x – 7 = 9. A. x = 2 B. x = 3 C. x = 4 D. x = 5 Answer: C Explanation: Add 7 to both sides to have 4x = 16, then divide by 4 to obtain x = 4. Question 10: Solve the equation: 3(2x – 1) = 5x + 4. A. x = 5
Question 15: Solve the equation: 3(x + 4) – 2(x – 1) = 2x + 10. A. x = 2 B. x = 3 C. x = 4 D. x = 5 Answer: C Explanation: Expand to obtain 3x + 12 – 2x + 2 = 2x + 10; simplifying gives x + 14 = 2x + 10, so x = 4. Question 16: Solve the inequality: – 4x + 7 ≥ 15. A. x ≥ – 2 B. x ≤ – 2 C. x ≥ 2 D. x ≤ 2 Answer: B Explanation: Subtract 7 to get – 4x ≥ 8; dividing by – 4 (and reversing the inequality) results in x ≤ – 2. Question 17: Solve for x: 9x + 6 = 3(4x + 2). A. x = 0 B. x = 1 C. x = 2 D. x = 3 Answer: A Explanation: Expand the right side to get 9x + 6 = 12x + 6; subtract 9x to obtain 6 = 3x + 6, then subtract 6 to find 0 = 3x, so x = 0. Question 18: Solve the inequality: (2x/3) – 1 ≥ 3. A. x ≥ 3 B. x ≥ 6 C. x ≥ 9 D. x ≥ 12 Answer: B Explanation: Add 1 to both sides to get 2x/3 ≥ 4; multiplying by 3 gives 2x ≥ 12, so dividing by 2 yields x ≥ 6. Question 19: Solve for x: 5(x – 3) = 2x + 7. A. x = 6 B. x = 22/ C. x = 7 D. x = 8
Answer: B Explanation: Expand to get 5x – 15 = 2x + 7; subtract 2x and add 15 to obtain 3x = 22, so x = 22/3. Question 20: Determine x from: 3 – 2x = 5x – 10. A. x = 13/ B. x = 10/ C. x = 7/ D. x = 2 Answer: A Explanation: Rearranging gives 3 + 10 = 5x + 2x, so 13 = 7x and x = 13/7. Question 21: For f(x) = 2x + 3, what is f(4)? A. 7 B. 9 C. 11 D. 13 Answer: C Explanation: Substitute 4 into f(x): 2(4) + 3 = 8 + 3 = 11. Question 22: If f(x) = x², what is the domain of f? A. x ≥ 0 B. All real numbers C. x > 0 D. x ≠ 0 Answer: B Explanation: Squaring is defined for every real number. Question 23: For f(x) = 1/(x – 2), what is the domain? A. All real numbers B. All real numbers except x = 2 C. x > 2 D. x < 2 Answer: B Explanation: The function is undefined when the denominator is zero; thus x ≠ 2. Question 24: If f(x) = x² and g(x) = x + 1, what is f(g(3))? A. 7 B. 8 C. 15 D. 16
Answer: B Explanation: f(0) = 0² – 1 = – 1. Question 30: If f(x) = x + 2 and g(x) = 3x, what is (f ∘ g)(x)? A. 3x + 2 B. 3x + 5 C. 2x + 3 D. 3x – 2 Answer: A Explanation: f(g(x)) = g(x) + 2 = 3x + 2. Question 31: For f(x) = |x|, what is f(–7)? A. – 7 B. 0 C. 7 D. 14 Answer: C Explanation: The absolute value of – 7 is 7. Question 32: What is the range of f(x) = 2x + 1? A. x ≥ 1 B. All real numbers C. x > 1 D. x < 1 Answer: B Explanation: A non-horizontal linear function covers all real numbers in its range. Question 33: If f(x) = x² + 2x + 1, what type of function is f? A. Linear B. Quadratic C. Cubic D. Exponential Answer: B Explanation: The function is quadratic and can be written as (x + 1)². Question 34: For f(x) = 1/x, what is the range? A. All real numbers B. All real numbers except 0 C. x > 0 D. x < 0
Answer: B Explanation: The function is never zero because 1/x ≠ 0 for any x. Question 35: If f(x) = √(x – 3), what is its domain? A. x > 3 B. x ≥ 3 C. All real numbers D. x < 3 Answer: B Explanation: The expression under the square root must be non-negative, so x ≥ 3. Question 36: What is the inverse of f(x) = (x – 1)/2? A. 2x – 1 B. (x + 1)/ C. 2x + 1 D. (x – 1)/ Answer: C Explanation: Write y = (x – 1)/2, solve for x: x = 2y + 1, hence f⁻¹(x) = 2x + 1. Question 37: For f(x) = |x – 2|, what is f(5)? A. 1 B. 2 C. 3 D. 4 Answer: C Explanation: f(5) = |5 – 2| = 3. Question 38: Determine the composition f(g(x)) if f(x) = x² and g(x) = x – 4. A. (x – 4)² B. x² – 4 C. x² – 8x + 16 D. Both A and C are correct Answer: D Explanation: f(g(x)) = (x – 4)², which expands to x² – 8x + 16. Question 39: If f(x) = 2x + 3 and g(x) = x², what is (g ∘ f)(2)? A. 25 B. 36 C. 49 D. 64
Answer: A Explanation: x² – 4x + 4 factors as (x – 2)². Question 45: Solve the equation: x² – 5x + 6 = 0. A. x = 1 or 6 B. x = 2 or 3 C. x = – 2 or – 3 D. x = 3 or 4 Answer: B Explanation: Factor as (x – 2)(x – 3) = 0, so x = 2 or 3. Question 46: Find the zeros of f(x) = x² – 4. A. x = 2 and – 2 B. x = 4 and – 4 C. x = 0 and 4 D. x = 1 and – 1 Answer: A Explanation: Factor as (x – 2)(x + 2) = 0. Question 47: Factor the sum/difference of cubes: 8x³ – 27. A. (2x – 3)(4x² + 6x + 9) B. (4x – 9)(2x² + 3x + 1) C. (2x + 3)(4x² – 6x + 9) D. (8x – 27)(x² + x + 1) Answer: A Explanation: Write 8x³ as (2x)³ and 27 as 3³; then use the formula a³ – b³ = (a – b)(a² + ab + b²). Question 48: Simplify the expression: (x³ – 27)/(x – 3). A. x² – 3x + 9 B. x² + 3x + 9 C. x² + 9x + 3 D. x² – 9 Answer: B Explanation: Factor x³ – 27 as (x – 3)(x² + 3x + 9) and cancel the common factor. Question 49: Factor completely: x³ + 3x² – 4x. A. x(x² + 3x – 4) B. x(x + 4)(x – 1) C. (x + 4)(x² – x) D. (x – 4)(x² + 3x)
Answer: B Explanation: Factor out x to get x(x² + 3x – 4), then factor the quadratic as (x + 4)(x – 1). Question 50: Solve the equation: x² – 16 = 0. A. x = 4 or – 4 B. x = 8 or – 8 C. x = 0 or 16 D. x = 2 or – 2 Answer: A Explanation: Recognize the difference of squares: (x – 4)(x + 4) = 0. Question 51: Find all real roots of f(x) = 2x³ – 3x² – 8x + 12 = 0. A. x = 2, 3/2, – 2 B. x = 2, – 3/2, – 2 C. x = – 2, – 3/2, 2 D. x = 2, 3, – 2 Answer: A Explanation: Testing x = 2 shows it is a root; factoring yields 2x² + x – 6, which factors as (2x – 3)(x + 2) = 0, so the roots are x = 2, 3/2, and – 2. Question 52: Factor completely: x⁴ – 16. A. (x² – 4)(x² + 4) B. (x – 2)(x + 2)(x² + 4) C. (x² – 4)² D. Both A and B Answer: D Explanation: x⁴ – 16 factors as a difference of squares: (x² – 4)(x² + 4), and x² – 4 further factors as (x – 2)(x + 2). Question 53: Factor by grouping: x³ + 2x² + 3x + 6. A. (x + 2)(x² + 3) B. (x + 3)(x² + 2) C. (x + 2)(x² + 3x + 3) D. (x + 1)(x² + x + 6) Answer: A Explanation: Group as (x³ + 2x²) + (3x + 6) = x²(x + 2) + 3(x + 2) = (x + 2)(x² + 3). Question 54: Solve the equation: x³ – x² – x + 1 = 0. A. x = 1 (triple root) B. x = 1 (double root) and x = – 1 C. x = 1 and x = – 1 D. x = – 1 only
Answer: A Explanation: The product of the roots is c/a = – 3/2. Question 60: Solve for x: x⁴ – 5x² + 4 = 0. A. x = ±1, ± B. x = ±1 only C. x = ±2 only D. x = 1, 2, 4, – 1 Answer: A Explanation: Let u = x², then u² – 5u + 4 = 0 factors as (u – 1)(u – 4) = 0, so x² = 1 or 4. Question 61: Simplify the rational expression: (x² – 9)/(x² – 6x + 9). A. (x + 3)/(x – 3) B. (x – 3)/(x + 3) C. (x + 3)²/(x – 3)² D. (x – 3)²/(x + 3)² Answer: A Explanation: Factor numerator as (x – 3)(x + 3) and denominator as (x – 3)², then cancel one (x – 3). Question 62: Simplify: (2/x) ÷ (4/x²). A. x/ B. 2/x C. x²/ D. 2x Answer: A Explanation: Division is equivalent to multiplying by the reciprocal: (2/x) · (x²/4) = x/2. Question 63: Add the rational expressions: 1/x + 2/x. A. 2/x B. 3/x C. 1/(2x) D. (2x + 1)/x Answer: B Explanation: The sum is (1 + 2)/x = 3/x. Question 64: Subtract: (3/(x + 1)) – (1/(x + 1)). A. 1/(x + 1) B. 2/(x + 1) C. (x + 1)/ D. 0
Answer: B Explanation: Since the denominators are the same, subtract the numerators: (3 – 1)/(x + 1) = 2/(x
- 1). Question 65: Multiply: (x/(x – 2)) · ((x + 3)/x). A. (x + 3)/(x – 2) B. x(x + 3)/(x – 2)² C. (x² + 3x)/(x(x – 2)) D. (x + 3)/(x) Answer: A Explanation: Cancel the common factor x to obtain (x + 3)/(x – 2). Question 66: Simplify: (x² – 4)/(x – 2). A. x – 2 B. x + 2 C. x² – 2 D. 1 Answer: B Explanation: Factor numerator as (x – 2)(x + 2) and cancel (x – 2). Question 67: Solve the equation: 1/(x – 1) + 1/(x + 1) = 1. A. x = 1 ± √ B. x = 1 + √2, 1 – √ C. x = 1 ± 2√ D. x = √ Answer: A Explanation: Combining the fractions gives (2x)/(x² – 1) = 1; cross-multiplying yields 2x = x² – 1, so x² – 2x – 1 = 0; solving with the quadratic formula gives x = 1 ± √2. Question 68: Solve for x: (x + 2)/(x – 3) = 2. A. x = 8 B. x = 4 C. x = 5 D. x = 6 Answer: A Explanation: Cross-multiply: x + 2 = 2(x – 3) leads to x + 2 = 2x – 6, so x = 8. Question 69: Simplify: (3x² – 9x)/(3x). A. x – 3 B. 3x – 9 C. x + 3 D. 3 – x
Question 74: Simplify: (4x² – 9)/(2x – 3). A. 2x + 3 B. 2x – 3 C. 4x + 9 D. 4x – 9 Answer: A Explanation: Factor numerator as (2x – 3)(2x + 3) and cancel (2x – 3). Question 75: Factor and simplify: (x² – 5x + 6)/(x² – 4). A. (x – 3)/(x + 2) B. (x – 2)/(x + 2) C. (x – 3)/(x – 2) D. (x – 2)/(x – 3) Answer: A Explanation: Factor numerator as (x – 2)(x – 3) and denominator as (x – 2)(x + 2); cancel (x – 2). Question 76: What is the common denominator of 1/(x – 1) and 1/(x + 2)? A. (x – 1)(x + 2) B. x – 1 + x + 2 C. (x – 1) + (x + 2) D. (x – 1)/(x + 2) Answer: A Explanation: The least common denominator is the product (x – 1)(x + 2). Question 77: Solve the equation: 2/(x + 1) – 3/(x – 1) = 0. A. x = – 5 B. x = 5 C. x = – 2 D. x = 2 Answer: A Explanation: Combine over a common denominator: [2(x – 1) – 3(x + 1)]/(x² – 1) = 0 leads to – x
- 5 = 0, so x = – 5. Question 78: Solve for x: (x/(x – 1)) + (1/(x – 1)) = 3. A. x = 1 B. x = 2 C. x = 3 D. x = 4 Answer: B Explanation: Combine numerator: (x + 1)/(x – 1) = 3; cross-multiply to get x + 1 = 3(x – 1), so x = 2.
Question 79: Simplify: (x² – 2x)/x. A. x – 2 B. x – 1 C. x² – 2 D. 1 – 2x Answer: A Explanation: Factor numerator as x(x – 2) and cancel x (x ≠ 0). Question 80: Simplify: ((x + 1)/(x – 1)) ÷ ((x² – 1)/(x + 1)). A. (x + 1)/(x – 1)² B. (x + 1)²/(x – 1)² C. (x – 1)/(x + 1) D. (x – 1)²/(x + 1) Answer: A Explanation: Recognize x² – 1 = (x + 1)(x – 1) and simplify accordingly. Question 81: Simplify: √49. A. 5 B. 6 C. 7 D. 8 Answer: C Explanation: √49 = 7. Question 82: Simplify: √(x²). A. x B. |x| C. x² D. 0 Answer: B Explanation: √(x²) equals the absolute value of x. Question 83: Simplify: √50. A. 5√ B. 2√ C. 25√ D. √ Answer: A Explanation: Since 50 = 25×2, √50 = 5√2.
Question 89: Simplify: √(a²b) for a, b ≥ 0. A. a√b B. √a·b C. ab D. √(ab) Answer: A Explanation: Factor out a² to get a√b. Question 90: Rationalize the denominator: 1/(√x + 1). A. (√x – 1)/(x – 1) B. (√x + 1)/(x – 1) C. (√x – 1)/(x + 1) D. (√x + 1)/(x + 1) Answer: A Explanation: Multiply by the conjugate (√x – 1) to obtain (√x – 1)/(x – 1). Question 91: Simplify: √50 – √8. A. 2√ B. 3√ C. 4√ D. 5√ Answer: B Explanation: √50 = 5√2 and √8 = 2√2; thus, the difference is 3√2. Question 92: Solve for x: √(x + 6) = x – 2. A. x = (5 + √33)/ B. x = (5 – √33)/ C. x = (5 + √33)/2 (only valid solution) D. x = (5 – √33)/2 (only valid solution) Answer: C Explanation: Squaring both sides gives x² – 5x – 2 = 0; only the positive solution x = (5 + √33)/ satisfies the original equation. Question 93: Simplify: (√a)/(√b). A. √(a·b) B. √(a/b) C. a/√b D. √(b/a) Answer: B Explanation: Division under a single radical gives √(a/b).
Question 94: If √x = 4, what is x? A. 4 B. 8 C. 12 D. 16 Answer: D Explanation: Squaring both sides, x = 16. Question 95: Simplify: √75. A. 3√ B. 5√ C. 5√ D. 15 Answer: B Explanation: Write 75 as 25×3; hence, √75 = 5√3. Question 96: Simplify: (√2)(√3). A. √ B. √ C. 5 D. 6 Answer: B Explanation: Multiply under the radical: √2·√3 = √6. Question 97: Solve: √(9x + 1) = 10. A. x = 11 B. x = 10 C. x = 9 D. x = 8 Answer: A Explanation: Square both sides: 9x + 1 = 100; subtract 1 and divide by 9 to get x = 11. Question 98: Rationalize the denominator: 3/(2 + √3). A. 3(2 + √3)/ (4 – 3) B. 3(2 – √3)/ (4 – 3) C. 3(2 – √3)/ (4 + 3) D. 3(2 + √3)/ (4 + 3) Answer: B Explanation: Multiply numerator and denominator by the conjugate (2 – √3) to obtain 3(2 – √3)/(4 – 3) = 3(2 – √3).
