Add, Subtract, Factor Linear Expressions - 7th Grade Math

A student writes two expressions for points scored in a game: $\left(\frac{3}{4}x+5\right)$ and $\left(\frac{1}{2}x-3\right)$ . Simplify the sum $\left(\frac{3}{4}x+5\right)+\left(\frac{1}{2}x-3\right)$ .
Answer: $\frac{5}{4}x+2$ This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, (3/4 x + 5) + (1/2 x - 3) involves combining like terms: first convert 1/2 to 2/4, so 3/4 x + 2/4 x = 5/4 x, and 5 + (-3) = 2, resulting in 5/4 x + 2. In this case, the correct simplification is (3/4 x + 5) + (1/2 x - 3) = 5/4 x + 2. A common error is forgetting to find a common denominator when adding fractions, such as adding 3/4 + 1/2 directly as 3/4 + 1/2 = 4/6 instead of 5/4. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
The expression $ax + bx + c$ can be factored as $x(a + b) + c$ when certain conditions are met. Which statement about this factoring is most accurate?
Answer: This factoring is always mathematically correct but may not represent the greatest common factor When you encounter factoring problems, remember that factoring means rewriting an expression as a product of simpler terms. The key is understanding when factoring is correct versus when it's complete. The expression $ax + bx + c$ can indeed be factored as $x(a + b) + c$ because you're pulling out the common factor $x$ from the first two terms. This is mathematically valid regardless of the values of $a$ , $b$ , or $c$ . However, this doesn't mean you've found the greatest common factor of all terms. Let's examine why the other options miss the mark. Choice A incorrectly claims this only works when $c = 0$ . While having $c = 0$ would allow complete factoring as $x(a + b)$ , the partial factoring $x(a + b) + c$ is still mathematically correct when $c \neq 0$ . Choice B gets the relationship backwards—when $c = 0$ , you can factor completely, making the factoring more complete, not less. Choice C focuses on the coefficient $a + b = 1$ , but this restriction isn't necessary for the factoring to be valid. Choice D correctly identifies that this factoring is always mathematically sound but acknowledges it may not represent the most simplified form. For instance, if all terms share a common factor, you haven't fully factored until you extract that greatest common factor. Study tip: When factoring, always check if you can pull out more common factors after your first step. Complete factoring means finding the greatest common factor of all terms, not just some of them.
Simplify the expression $0.5x+2.5-0.3x+1.5$ by combining like terms.
Answer: $0.2x+4$ This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, 0.5x + 2.5 - 0.3x + 1.5 combines x terms (0.5 - 0.3 = 0.2x) and constants (2.5 + 1.5 = 4), giving 0.2x + 4. In this case, the correct simplification is 0.2x + 4. A common error is combining unlike terms or mishandling decimals, such as adding all numbers without grouping. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
When factoring $21a - 14b + 35c$ , a student writes $7(3a - 2b + 5c)$ . What should the student do to verify this factorization is correct?
Answer: Use the distributive property to multiply $7(3a - 2b + 5c)$ and compare to the original To verify any factorization, expand it using the distributive property: $7(3a - 2b + 5c) = 21a - 14b + 35c$ , which matches the original expression. While choices A, B, and D describe properties of correct factorization, choice C describes the direct verification method that confirms the factorization is mathematically equivalent to the original expression.
A coach writes an expression for total practice time: $3\left(\frac{2}{3}x-5\right)$ . Expand the expression using the distributive property.
Answer: $2x-15$ This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations include adding by combining like terms (3x+2x=5x, coefficients add), subtracting by distributing the negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expanding using the distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), and factoring by finding the GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires the same variable (3x and 5x combine, but 2x and 3 don't). For example, (3x+5)+(2x-3) combines like terms: 3x+2x=5x, 5+(-3)=2, giving 5x+2; or (4x+7)-(2x+3) distributes the negative: 4x+7-2x-3, combines to 2x+4; or factor 8x+12: GCF=4, so 4(2x+3). Expand by distributing 3: 3*(2/3)x = 2x, and 3*(-5) = -15, resulting in 2x - 15, which is choice A. A common error is incomplete distribution, like multiplying only the first term and forgetting the second, giving just 2x without -15. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
Factor the expression $\frac{3}{5}x+\frac{6}{5}$ by taking out the greatest common factor.
Answer: $\frac{3}{5}(x+2)$ This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, to factor 3/5 x + 6/5, find the GCF of 3/5 and 6/5 which is 3/5, then 3/5 (x + 2). In this case, the correct factoring is 3/5 (x + 2). A common error is not identifying the full GCF for fractions or factoring incompletely, such as using 6/5 instead. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
Combine like terms to simplify $\left(2x-\frac{3}{4}\right)+\left(-5x+\frac{1}{2}\right)$ .
Answer: $-3x-\frac{1}{4}$ This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, (2x - 3/4) + (-5x + 1/2) combines x terms (2 - 5 = -3x) and constants (-3/4 + 2/4 = -1/4), giving -3x - 1/4. In this case, the correct simplification is -3x - 1/4. A common error is sign error in adding fractions without a common denominator, such as -3/4 + 1/2 = -3/4 + 1/2 = 1/4 instead of -1/4. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
In a science lab, the temperature change is modeled by subtracting two expressions. Simplify the difference: (4.2x-1.5)-(1.8x+2.5).
Answer: 2.4x-4.0 Distributing the negative sign across the second expression gives 4.2x - 1.5 - 1.8x - 2.5. Combining the x-terms: 4.2x - 1.8x = 2.4x. Combining the constants: -1.5 - 2.5 = -4.0. Putting these together gives 2.4x - 4.0, matching choice C. A common error is mishandling the constants' signs, such as computing -1.5 + 2.5 = 1.0 instead of -1.5 - 2.5 = -4.0, which leads to choice B.
Expand the expression $-3\left(2x-\frac{5}{3}\right)$ using the distributive property.
Answer: $-6x+5$ This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations include adding by combining like terms (3x+2x=5x, coefficients add), subtracting by distributing the negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expanding using the distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), and factoring by finding the GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires the same variable (3x and 5x combine, but 2x and 3 don't). A specific example is expanding 3(2x-5)=6x-15 by distributing to each term. For this problem, distribute -3 to each term: -3*2x = -6x, -3*( $-(\frac{5}{3})$ ) = +5, resulting in -6x+5. A common error is incomplete distribution, like missing the second term or mishandling the negative sign with the fraction. Strategy: for expanding, distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); common mistakes include distributing negative to only the first term -(2x-3)=-2x-3 (wrong, should be -2x+3), or fraction operations without proper multiplication.
Expand the expression for total cost: $-2\left(3x-\frac{7}{2}\right)$ .
Answer: $-6x+7$ This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations include adding by combining like terms (3x+2x=5x, coefficients add), subtracting by distributing the negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expanding using the distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), and factoring by finding the GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires the same variable (3x and 5x combine, but 2x and 3 don't). For example, (3x+5)+(2x-3) combines like terms: 3x+2x=5x, 5+(-3)=2, giving 5x+2; or (4x+7)-(2x+3) distributes the negative: 4x+7-2x-3, combines to 2x+4; or factor 8x+12: GCF=4, so 4(2x+3). Expand by distributing -2: -2*3x = -6x, -2*(-7/2) = +7, giving -6x + 7, which is choice B. A common error is sign error in distribution, like -2*(-7/2) as -7 instead of +7. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
A student factored the expression $12x + 18y - 6$ and wrote $6(2x + 3y - 1)$ . To verify this is correct, which property should be applied?
Answer: The distributive property to multiply $6$ by each term inside the parentheses To verify factoring is correct, use the distributive property to expand $6(2x + 3y - 1) = 12x + 18y - 6$ , confirming it matches the original expression. Choice A (commutative) deals with order, choice B (associative) deals with grouping, and choice D (identity) is about value preservation but doesn't describe the verification method.
Factor the expression $8x+12$ completely by finding the greatest common factor (GCF).
Answer: $4(2x+3)$ This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations include adding by combining like terms (3x+2x=5x, coefficients add), subtracting by distributing the negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expanding using the distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), and factoring by finding the GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires the same variable (3x and 5x combine, but 2x and 3 don't). A specific example is factoring 8x+12: GCF=4, so 4(2x+3). For this problem, the GCF of 8x and 12 is 4, divide each by 4: 8x/4=2x, 12/4=3, giving 4(2x+3). A common error is factoring partially, like pulling out only 2 instead of the full GCF 4, resulting in 2(4x+6). Strategy: for factoring, find GCF of all terms, divide each term by GCF, write as GCF(quotients); common mistakes include factoring incompletely.
Sara needs to add $3(x - 2) + 5x$ and $2(3 - x) - 4$ . If she first expands each expression completely before combining, what will be her final simplified answer?
Answer: $6x - 4$ When you encounter expressions that need to be added together, the key is to expand each expression first using the distributive property, then combine like terms systematically. Let's expand the first expression: $3(x - 2) + 5x$ . Using the distributive property, $3(x - 2) = 3x - 6$ , so this becomes $3x - 6 + 5x = 8x - 6$ . For the second expression: $2(3 - x) - 4$ . Distributing gives us $2(3 - x) = 6 - 2x$ , so this becomes $6 - 2x - 4 = 2 - 2x$ . Now we add the expanded expressions: $(8x - 6) + (2 - 2x)$ . Combining like terms: $8x - 2x - 6 + 2 = 6x - 4$ . Looking at the wrong answers: Choice A ( $6x - 6$ ) likely comes from incorrectly combining the constant terms as $-6 - 4 = -10$ instead of $-6 + 2 = -4$ . Choice B ( $6x + 2$ ) results from sign errors when distributing or combining terms. Choice C ( $8x - 4$ ) happens when you correctly handle the constants but forget to combine the $x$ terms: $8x - 2x = 6x$ , not $8x$ . The correct answer is D: $6x - 4$ . Strategy tip: Always expand completely before combining, and double-check your signs when distributing negative terms. Work step-by-step rather than trying to do multiple operations mentally—this prevents the sign errors that create most wrong answers in these problems.
Factor the expression $15x-10$ completely by finding the greatest common factor (GCF).
Answer: $5(3x-2)$ This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations include adding by combining like terms (3x+2x=5x, coefficients add), subtracting by distributing the negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expanding using the distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), and factoring by finding the GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires the same variable (3x and 5x combine, but 2x and 3 don't). A specific example is factoring 8x+12: GCF=4, so 4(2x+3). For this problem, the GCF of 15x and 10 is 5, divide each by 5: 15x/5=3x, 10/5=2, but since it's 15x-10, it's 5(3x-2). A common error is factoring partially, like pulling out only 1 or ignoring the negative sign. Strategy: for factoring, find GCF of all terms, divide each term by GCF, write as GCF(quotients); common mistakes include factoring incompletely or mishandling signs.
In science class, the temperature change is modeled by $\left(4x+7\right)-\left(2x+3\right)$ . Simplify the difference.
Answer: $2x+4$ This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, (4x+7)-(2x+3) requires distributing the negative: 4x+7-2x-3, then combine like terms: 4x-2x=2x, 7-3=4, giving 2x+4. In this case, the correct simplification is (4x+7)-(2x+3)=2x+4. A common error is not distributing the negative to all terms in the subtracted expression, such as treating it as 4x+7-2x+3 instead of -3. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
A coach says the total practice time can be written as $3(2x-5)$ . Expand the expression using the distributive property.
Answer: $6x-15$ This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, expanding 3(2x-5) using the distributive property gives 3*2x - 3*5 = 6x - 15. In this case, the correct expansion is 3(2x-5)=6x-15. A common error is missing the distribution to the second term or mishandling the sign, such as 3*2x -5 =6x-5 instead of multiplying the 5 by 3. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
During basketball practice, the coach compares two drills. The number of shots is modeled by (4x+7) and (2x-3). Find and simplify the difference (4x+7)-(2x-3).
Answer: 2x+10 Distribute the negative across the second expression: 4x + 7 - 2x + 3 (since -(2x-3) = -2x+3). Combine the x-terms: 4x - 2x = 2x. Combine the constants: 7 + 3 = 10. Putting these together gives 2x + 10, matching choice D. Choice B (2x+4) comes from only partially distributing the negative, treating -(2x-3) as -2x-3 instead of -2x+3, which flips the sign of the constant. Choices A (2x-10) and C (6x+4) involve further sign or combination errors in the subtraction.
Combine like terms to simplify $\left(2x-\frac{3}{4}\right)+\left(-5x+\frac{1}{2}\right)$ .
Answer: $-3x-\frac{1}{4}$ This question tests adding, subtracting, factoring, and expanding linear expressions with rational coefficients using properties of operations. Operations: add by combining like terms (3x+2x=5x, coefficients add), subtract by distributing negative then combining (4x-(2x-3)=4x-2x+3=2x+3, negative distributes to all terms), expand using distributive property a(b+c)=ab+ac (multiply each term: 3(2x-5)=6x-15), factor by finding GCF and dividing out (6x+9: GCF=3, so 3(6x/3+9/3)=3(2x+3)). Combining like terms requires same variable (3x and 5x combine, but 2x and 3 don't). For example, (2x - 3/4) + (-5x + 1/2) combines x terms (2 - 5 = -3x) and constants (-3/4 + 2/4 = -1/4), giving -3x - 1/4. In this case, the correct simplification is -3x - 1/4. A common error is sign error in adding fractions without a common denominator, such as -3/4 + 1/2 = -3/4 + 1/2 = 1/4 instead of -1/4. Strategy: (1) for adding/subtracting: distribute any negatives first (important for subtraction), identify like terms (same variable part: 3x and 5x are like, 2x and 3 aren't), combine (add/subtract coefficients: 3x+5x=8x), combine constants separately (5-3=2); (2) for expanding: distribute to every term (a(b+c+d)=ab+ac+ad, don't miss any); (3) for factoring: find GCF of all terms, divide each term by GCF, write as GCF(quotients). Common mistakes: distributing negative to only first term -(2x-3)=-2x-3 (wrong, should be -2x+3), combining unlike terms, fraction operations without common denominators, factoring incompletely.
Factor the linear expression completely: 9x-12.
Answer: 3(3x-4) The GCF of 9x and -12 is 3. Dividing each term by 3 gives 9x/3 = 3x and -12/3 = -4, so the factored form is 3(3x - 4), matching choice A. Choice B (-3(3x+4)) uses the wrong sign for the GCF, which flips the signs inside the parentheses incorrectly. Choice C uses 6, which isn't the GCF of 9 and 12, so the x-coefficient becomes a fraction (3/2) instead of a whole number. Choice D doesn't divide -12 by 9 at all: 9(x-12) actually equals 9x - 108, not the original expression.
A student records two parts of a science lab time as expressions: (3/4 x + 2) minutes and (1/2 x - 5/2) minutes. Simplify the sum (3/4 x + 2) + (1/2 x - 5/2).
Answer: 5/4 x - 1/2 Combine the x-terms: 3/4 x + 1/2 x = 3/4 x + 2/4 x = 5/4 x. Combine the constants: 2 - 5/2 = 4/2 - 5/2 = -1/2. Adding these results gives 5/4 x - 1/2, matching choice D. A common mistake is adding 3/4 and 1/2 without a common denominator, which can lead to an answer like 5/6 x instead of the correct 5/4 x.

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