\(\frac{3}{4}(2\mathrm{x} + 8) - \frac{1}{3}(6\mathrm{x} - 9)\)Which of the following is equivalent to the expression above?

1. INFER the solution approach

• Given: \(\frac{3}{4}(2x + 8) - \frac{1}{3}(6x - 9)\)
• Strategy: Apply the distributive property to both terms, then combine like terms
• Key insight: Handle each parenthetical expression separately first

2. SIMPLIFY the first term using distributive property

• \(\frac{3}{4}(2x + 8) = \frac{3}{4} \cdot 2x + \frac{3}{4} \cdot 8\)
• \(= \frac{6x}{4} + \frac{24}{4}\)
• \(= \frac{3x}{2} + 6\)

3. SIMPLIFY the second term using distributive property

• \(-\frac{1}{3}(6x - 9) = -\frac{1}{3} \cdot 6x - \frac{1}{3} \cdot (-9)\)
• \(= -\frac{6x}{3} + \frac{9}{3}\)
• \(= -2x + 3\)
• Watch the signs carefully: \(-\frac{1}{3}\) times \((-9)\) gives positive 3

4. SIMPLIFY by combining the results

• \(\frac{3x}{2} + 6 - 2x + 3\)
• Group like terms: \((\frac{3x}{2} - 2x) + (6 + 3)\)

5. SIMPLIFY the x terms

• \(\frac{3x}{2} - 2x = \frac{3x}{2} - \frac{4x}{2} = -\frac{x}{2} = -\frac{1}{2}x\)
• Constants: \(6 + 3 = 9\)

Answer: (B) \(-\frac{1}{2}x + 9\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution with signs: When distributing \(-\frac{1}{3}(6x - 9)\), students often incorrectly handle \(-\frac{1}{3} \cdot (-9)\), getting \(-3\) instead of \(+3\).

This gives them \(-\frac{1}{2}x + 6\) instead of \(-\frac{1}{2}x + 9\), leading to confusion since this exact answer isn't among the choices, causing them to guess or select Choice (A) \(-\frac{1}{2}x + 3\) as the closest match.

Second Most Common Error:

Poor SIMPLIFY technique with fractions: Students struggle with \(\frac{3x}{2} - 2x\), either not converting \(2x\) to \(\frac{4x}{2}\) or making arithmetic errors in the subtraction.

Common mistake: \(\frac{3x}{2} - 2x = \frac{1x}{2}\) instead of \(-\frac{x}{2}\), leading them to select Choice (D) \(\frac{1}{2}x + 9\) with the wrong sign on the x coefficient.

The Bottom Line:

This problem tests systematic algebraic manipulation under pressure. Students who rush through the distributive property or fraction arithmetic typically select incorrect answers, while those who carefully track signs and convert to common denominators succeed.