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ACT Math Help: Equivalent Expressions

Review real example questions for Equivalent Expressions in ACT Math.

Question 1 / 10

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

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Question 1

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

$5x - 4$
$5x - 12$ (correct answer)
$6x - 12$
$x - 12$

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 2

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

$7x^3y^6$
$12x^3y^8$
$12x^4y^6$ (correct answer)
$7x^4y^6$

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 3

A student rewrites an expression while simplifying. What is the simplified form of $7x - 3 + 2x + 5$ ?

$9x - 2$
$9x - 8$
$9x + 2$ (correct answer)
$5x + 2$

Explanation: Simplifying means combining like terms, and each term travels with the sign in front of it. The $x$ terms give $7x + 2x = 9x$ , and the constants give $-3 + 5 = 2$ , so the expression simplifies to $9x + 2$ . The answer $9x - 2$ subtracts the constants as $3 - 5$ instead of adding $-3$ and $5$ ; $9x - 8$ treats the $+5$ as $-5$ and computes $-3 - 5$ ; and $5x + 2$ subtracts the variable terms as $7x - 2x$ even though the $2x$ is being added. Before combining, rewrite the expression as a sum such as $7x + (-3) + 2x + 5$ so every sign is attached to its own term and cannot drift.

Question 4

Which expression is equivalent to $4x(2 + 3)$ ?

$10$
$20x$ (correct answer)
$10x$
$20$

Explanation: Order of operations says to simplify inside the parentheses before multiplying, and the variable outside stays attached through the multiplication. Inside, $2 + 3 = 5$ , so the expression becomes $4x \cdot 5 = 20x$ ; distributing gives the same result, since $4x(2) + 4x(3) = 8x + 12x = 20x$ . The value $20$ carries out the multiplication correctly but drops the variable, which cannot disappear because nothing cancels it. Both $10x$ and $10$ come from multiplying the 5 by the 2 inside the parentheses instead of by the 4 outside, and $10$ additionally loses the $x$ . Whenever a coefficient with a variable multiplies a numerical sum, collapse the sum first and keep the variable on the product.

Question 5

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

$6x+15$
$21x$ (correct answer)
$7x$
$21$

Explanation: Simplify inside the parentheses first, then multiply, keeping the variable attached to the product. Since $2 + 5 = 7$ , the expression becomes $3x \cdot 7 = 21x$ , and distributing agrees, because $3x(2) + 3x(5) = 6x + 15x = 21x$ . The value $21$ does the multiplication correctly but loses the $x$ , which nothing in the problem cancels. The expression $6x+15$ distributes but drops the variable from the second product, since $3x(5)$ is $15x$ , not $15$ , and $7x$ comes from adding inside the parentheses and then forgetting to multiply by the 3. When a term like $3x$ multiplies a numerical sum, every piece of the product keeps the variable, so a variable-free term signals a lost factor.

Question 6

Simplify: $6(m - 2) + 4(m + 1)$

$10m - 2$
$10m + 2$
$10m - 8$ (correct answer)
$10m + 10$

Explanation: To simplify this expression, we need to apply the distributive property to both terms. First, distribute: 6(m - 2) = 6m - 12 and 4(m + 1) = 4m + 4. The expression becomes 6m - 12 + 4m + 4. Combining like terms: (6m + 4m) + (-12 + 4) = 10m - 8. Choice B incorrectly has +2 instead of -8 as the constant term.

Question 7

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

$7x - 6x + 12$
$x + 12$ (correct answer)
$13x - 12$
$x - 12$

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 $2x^2(3x)$ ?

$6x^6$
$6x^5$
$5x^2$
$6x^3$ (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 9

Which of the following expressions is equivalent to $4(a + 2b) - 2(a - 3b)$ ?

$2a + 2b$
$2a + 5b$
$6a + 14b$
$2a + 14b$ (correct answer)

Explanation: This is a distribution and simplification question testing the distributive property with negatives. Choice D (2a + 14b) is correct — distribute the 4: 4a + 8b. Distribute the −2 across (a − 3b): −2a + 6b. Note: −2 × (−3b) = +6b, not −6b. Combine like terms: (4a − 2a) + (8b + 6b) = 2a + 14b. Choice A (2a + 2b) results from treating the second distribution as −2(a − 3b) = −2a − 6b (wrong sign on 3b), giving 8b − 6b = 2b. Choice B (2a + 5b) is an arithmetic error in combining the b terms, possibly computing 8b − 3b. Choice C (6a + 14b) adds 4a + 2a = 6a instead of subtracting, getting the sign wrong on the a-coefficient of the second term. Pro tip: When distributing a negative number, BOTH terms inside the parentheses change sign. Write out −2(a − 3b) = −2a + 6b before combining anything.

Question 10

Factor: $x^2 - 4x - 5$

$(x + 5)(x + 1)$
$(x - 1)(x + 5)$
$(x - 5)(x - 1)$
$(x - 5)(x + 1)$ (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.