Equivalent Expressions - ACT Math

Simplify: $2x - (4x - 6)$
Answer: $-2x + 6$ Distribute the negative sign to both terms in the parentheses: 2x - (4x - 6) = 2x - 4x + 6. Combine like terms by grouping x terms and constants: (2x - 4x) + 6 = -2x + 6. The simplified form is -2x + 6.
Simplify by combining like terms: $9a - 4(a - 2)$ .
Answer: $5a + 8$ First distribute -4 through the parentheses: -4(a - 2) = -4a + 8. Then combine with 9a: 9a - 4a + 8 = 5a + 8. Combine like terms: 9a - 4a = 5a, and the constant term is +8. Choice D incorrectly shows -8 instead of +8 for the constant.
Factor: $6m + 9$ .
Answer: $3(2m + 3)$ To factor this expression, find the greatest common factor of 6m and 9. Both terms share a factor of 3: $6m = 3(2m)$ and $9 = 3(3)$ . Therefore, $6m + 9 = 3(2m + 3)$ . Choice B factors out 6 incorrectly, while choice A has the wrong coefficients inside the parentheses.
Simplify: $2(x + 4) - 3(x - 1)$
Answer: $-x + 11$ To simplify this expression, we need to apply the distributive property and then combine like terms. First, distribute: $2(x + 4) = 2x + 8$ and $3(x - 1) = 3x - 3$ . The expression becomes $2x + 8 - 3x + 3$ . Combining like terms: $(2x - 3x) + (8 + 3) = -x + 11$ . Choice A incorrectly leaves the expression in expanded form without combining like terms.
Which expression is equivalent to $2x^2(3x)$ ?
Answer: $6x^3$ 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.
What is the simplified form of $6z + 2(3 - z)$ ?
Answer: $4z + 6$ To simplify this expression, we need to apply the distributive property. First, distribute the 2 to both terms in the parentheses: 2(3 - z) = 6 - 2z. The expression becomes 6z + 6 - 2z. Combining like terms: (6z - 2z) + 6 = 4z + 6. Choice D incorrectly has -6 instead of +6 as the constant term.
Which expression is equivalent to $2x(3 + 4)$ ?
Answer: $14x$ To simplify this expression, we need to first simplify inside the parentheses, then apply the distributive property. Inside the parentheses: 3 + 4 = 7. The expression becomes 2x(7) = 14x. We can also think of this as distributing: 2x(3 + 4) = 2x(3) + 2x(4) = 6x + 8x = 14x. Choice A incorrectly leaves the expression in distributed form without combining like terms.
A student rewrites the expression $3(2x-5)+4x$ to combine like terms. What is the simplified form of $3(2x-5)+4x$ ?
Answer: $10x-15$ To simplify this expression, we need to apply the distributive property and then combine like terms. First, distribute 3 to both terms inside the parentheses: 3(2x - 5) = 6x - 15. Now we have 6x - 15 + 4x, and we can combine the like terms 6x and 4x to get 10x. The final simplified form is 10x - 15. Choice A incorrectly shows 10x - 5, which would result from incorrectly distributing 3 × (-5) = -5 instead of -15.
Factor: $x^2 + x - 6$
Answer: $(x + 3)(x - 2)$ To factor this quadratic expression, we need to find two numbers that multiply to -6 and add to 1. The numbers 3 and -2 satisfy these conditions: 3 × (-2) = -6 and 3 + (-2) = 1. Therefore, x² + x - 6 = (x + 3)(x - 2). We can verify by expanding: (x + 3)(x - 2) = x² - 2x + 3x - 6 = x² + x - 6. Choice B incorrectly reverses the signs in the factors.
What is the simplified form of $3x + 5x - 2x$ ?
Answer: $6x$ To simplify this expression, combine like terms by adding and subtracting the coefficients of x. Start with 3x + 5x - 2x and group the x terms together: (3 + 5 - 2)x. Calculate the coefficients: 3 + 5 - 2 = 6. Therefore, the simplified form is 6x. Choice B incorrectly added all coefficients without considering the subtraction.
Which expression is equivalent to $3x(2 + 5)$ ?
Answer: $21x$ 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.
A student rewrites an expression while simplifying. What is the simplified form of $7x - 3 + 2x + 5$ ?
Answer: $9x + 2$ 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.
Simplify by combining like terms: $2x^2 + 3x - x^2 + 5x$ .
Answer: $x^2 + 8x$ Combine like terms by grouping x² terms and x terms separately. For x² terms: 2x² - x² = x². For x terms: 3x + 5x = 8x. Therefore, 2x² + 3x - x² + 5x = x² + 8x. Cannot combine x² and x terms since they are not like terms. Choice C incorrectly keeps the 2x² term.
What is the simplified form of $8a - 2(3a - 4)$ ?
Answer: $2a + 8$ To simplify this expression, we need to apply the distributive property. First, distribute the -2 to both terms in the parentheses: -2(3a - 4) = -6a + 8. The expression becomes 8a - 6a + 8. Combining like terms: (8a - 6a) + 8 = 2a + 8. Choice D incorrectly has -8 instead of +8 as the constant term.
Factor: $x^2 - 9x + 20$
Answer: $(x - 4)(x - 5)$ To factor this quadratic expression, we need to find two numbers that multiply to 20 and add to -9. The numbers -4 and -5 satisfy these conditions: (-4) × (-5) = 20 and (-4) + (-5) = -9. Therefore, x² - 9x + 20 = (x - 4)(x - 5). We can verify by expanding: (x - 4)(x - 5) = x² - 5x - 4x + 20 = x² - 9x + 20. Choice B incorrectly has positive factors instead of negative ones.
What is the simplified form of $5x + 3(2 - x)$ ?
Answer: $2x + 6$ To simplify this expression, we need to apply the distributive property. First, distribute the 3 to both terms in the parentheses: 3(2 - x) = 6 - 3x. The expression becomes 5x + 6 - 3x. Combining like terms: (5x - 3x) + 6 = 2x + 6. Choice D incorrectly has -6 instead of +6 as the constant term.
Factor: $x^2 + 2x - 8$
Answer: $(x + 4)(x - 2)$ To factor this quadratic expression, we need to find two numbers that multiply to -8 and add to 2. The numbers 4 and -2 satisfy these conditions: 4 × (-2) = -8 and 4 + (-2) = 2. Therefore, x² + 2x - 8 = (x + 4)(x - 2). We can verify by expanding: (x + 4)(x - 2) = x² - 2x + 4x - 8 = x² + 2x - 8. Choice B incorrectly reverses the signs in the factors.
Simplify: $8 - 3(2r + 1)$ .
Answer: $5 - 6r$ Apply the distributive property: -3(2r + 1) = -6r - 3. Then combine with the constant: 8 - 6r - 3 = 5 - 6r. Combine the constant terms: 8 - 3 = 5, while the -6r term remains unchanged. Choice D incorrectly adds the constants as 8 + 3 = 11.
Factor: $12k - 8$ .
Answer: $4(3k - 2)$ Find the greatest common factor of 12k and 8. Both terms share a factor of 4: 12k = 4(3k) and 8 = 4(2). Therefore, 12k - 8 = 4(3k - 2). Choice B factors out 2 instead of 4, while choice C has the wrong sign inside the parentheses.
Which of the following matches the expansion of $(x - 3)(x + 2)$ ?
Answer: $x^2 - x - 6$ To expand this expression, we need to apply the FOIL method. First: x × x = x². Outer: x × 2 = 2x. Inner: (-3) × x = -3x. Last: (-3) × 2 = -6. Combining all terms: x² + 2x - 3x - 6 = x² - x - 6. Choice B incorrectly has +5x instead of -x as the middle term.

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