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ACT Math · Learn by Concept

ACT Math Help: Equivalent Expressions

Review real example questions for Equivalent Expressions in ACT Math.

Question 1 / 10

0 of 10 answered

Simplify: 8−(2x−3)8 - (2x - 3)8−(2x−3)

Select an answer to continue

All questions

Question 1

Simplify: 8−(2x−3)8 - (2x - 3)8−(2x−3)

  1. 8−2x+38 - 2x + 38−2x+3
  2. 5−2x5 - 2x5−2x
  3. 11−2x11 - 2x11−2x (correct answer)
  4. 2x+112x + 112x+11

Explanation: Distribute the negative sign to both terms in the parentheses: 8 - (2x - 3) = 8 - 2x + 3. Combine the constant terms: 8 + 3 = 11, giving 8 - 2x + 3 = 11 - 2x. The simplified form is 11 - 2x, which equals -2x + 11.

Question 2

Which of the following expressions is equivalent to 3(x−4)+2x3(x - 4) + 2x3(x−4)+2x?

  1. 5x−45x - 45x−4
  2. 5x−125x - 125x−12 (correct answer)
  3. 6x−126x - 126x−12
  4. x−12x - 12x−12

Explanation: The correct answer is B (5x − 12). Distribute the 3 across the parentheses: 3(x − 4) = 3x − 12. Then combine like terms with 2x: 3x − 12 + 2x = 5x − 12. A (5x − 4) results from distributing 3 to x but not to −4, keeping −4 instead of computing 3 × (−4) = −12. C (6x − 12) comes from incorrectly treating the 2x as adding to the coefficient 3 rather than to 3x. D (x − 12) results from subtracting 2x rather than adding it. Distribution errors are extremely common — always multiply the outside factor by every term inside the parentheses.

Question 3

For all xxx and yyy, which of the following expressions is equivalent to (3x3y2)(4xy4)(3x^3y^2)(4xy^4)(3x3y2)(4xy4)?

  1. 7x3y67x^3y^67x3y6
  2. 12x3y812x^3y^812x3y8
  3. 12x4y612x^4y^612x4y6 (correct answer)
  4. 7x4y67x^4y^67x4y6

Explanation: The correct answer is C (12x⁴y⁶). Multiply the coefficients: 3 × 4 = 12. Apply the product rule for exponents (add exponents of like bases): x³ × x¹ = x^(3+1) = x⁴; y² × y⁴ = y^(2+4) = y⁶. Result: 12x⁴y⁶. A (7x³y⁶) adds coefficients (3+4=7) instead of multiplying and doesn't add the x exponent. B (12x³y⁸) correctly multiplies coefficients but treats x³ as fixed and incorrectly adds the y exponents (possibly counting y⁴ twice). D (7x⁴y⁶) adds coefficients but correctly adds the exponents. Key rule: multiply coefficients, add exponents of like bases.

Question 4

Which expression is equivalent to (x+2)(x−2)(x + 2)(x - 2)(x+2)(x−2)?

  1. x2−2x^2 - 2x2−2
  2. x2+4x^2 + 4x2+4
  3. x2−4x^2 - 4x2−4 (correct answer)
  4. 2x2−42x^2 - 42x2−4

Explanation: This is a difference of squares pattern: (a + b)(a - b) = a² - b². Apply this with a = x and b = 2: (x + 2)(x - 2) = x² - 2² = x² - 4. The middle terms cancel when expanding: x² - 2x + 2x - 4 = x² - 4. Choice B incorrectly shows +4 instead of -4.

Question 5

Which expression is equivalent to 3x(2+5)3x(2 + 5)3x(2+5)?

  1. 6x+15x6x + 15x6x+15x
  2. 21x21x21x (correct answer)
  3. 7x7x7x
  4. 212121

Explanation: To simplify this expression, we need to first simplify inside the parentheses, then apply the distributive property. Inside the parentheses: 2 + 5 = 7. The expression becomes 3x(7) = 21x. We can also think of this as distributing: 3x(2 + 5) = 3x(2) + 3x(5) = 6x + 15x = 21x. Choice A incorrectly leaves the expression in distributed form without combining like terms.

Question 6

Which of the following matches the expansion of −3(2x−5)-3(2x - 5)−3(2x−5)?

  1. −6x−15-6x - 15−6x−15
  2. −6x+15-6x + 15−6x+15 (correct answer)
  3. 6x−156x - 156x−15
  4. 6x+156x + 156x+15

Explanation: Apply the distributive property with -3 multiplied by each term inside the parentheses. -3 × 2x = -6x and -3 × (-5) = +15 (negative times negative equals positive). Therefore, -3(2x - 5) = -6x + 15. Choice A incorrectly shows -15 instead of +15.

Question 7

What is the simplified form of 7x−3(2x−4)7x - 3(2x - 4)7x−3(2x−4)?

  1. 7x−6x+127x - 6x + 127x−6x+12
  2. x+12x + 12x+12 (correct answer)
  3. 13x−1213x - 1213x−12
  4. x−12x - 12x−12

Explanation: To simplify this expression, we need to apply the distributive property. First, distribute the -3 to both terms in the parentheses: -3(2x - 4) = -6x + 12. The expression becomes 7x - 6x + 12. Combining like terms: (7x - 6x) + 12 = x + 12. Choice C incorrectly adds the coefficients of x terms instead of subtracting.

Question 8

Which expression is equivalent to x(x−7)+3xx(x - 7) + 3xx(x−7)+3x?

  1. x2−4xx^2 - 4xx2−4x (correct answer)
  2. x2−10xx^2 - 10xx2−10x
  3. x2+4xx^2 + 4xx2+4x
  4. x2−7x^2 - 7x2−7

Explanation: First distribute x through the parentheses: x(x - 7) = x² - 7x. Then add 3x to get: x² - 7x + 3x = x² - 4x. Combine like terms: -7x + 3x = -4x, while the x² term remains unchanged. Choice B incorrectly shows -10x instead of -4x.

Question 9

Which expression is equivalent to 2x2(3x)2x^2(3x)2x2(3x)?

  1. 6x66x^66x6
  2. 6x56x^56x5
  3. 5x25x^25x2
  4. 6x36x^36x3 (correct answer)

Explanation: Multiply the coefficients and add the exponents when multiplying powers with the same base. 2x² × 3x = (2 × 3)(x² × x) = 6x³. The exponents add: x² × x¹ = x²⁺¹ = x³. Choice A incorrectly multiplies the exponents instead of adding them.

Question 10

Factor: x2−4x−5x^2 - 4x - 5x2−4x−5

  1. (x+5)(x+1)(x + 5)(x + 1)(x+5)(x+1)
  2. (x−1)(x+5)(x - 1)(x + 5)(x−1)(x+5)
  3. (x−5)(x−1)(x - 5)(x - 1)(x−5)(x−1)
  4. (x−5)(x+1)(x - 5)(x + 1)(x−5)(x+1) (correct answer)

Explanation: To factor this quadratic expression, find two numbers that multiply to -5 (the constant term) and add to -4 (the coefficient of x). The numbers -5 and 1 satisfy both conditions: (-5) × 1 = -5 and (-5) + 1 = -4. Therefore, x² - 4x - 5 = (x - 5)(x + 1). You can verify by expanding: (x - 5)(x + 1) = x² + x - 5x - 5 = x² - 4x - 5.