\((\frac{1}{2}\mathrm{x} + 3) - (\frac{2}{3}\mathrm{x} - 5)\) Which of the following is equivalent to the expression above?...

1. TRANSLATE the problem information

• Given expression: \(\left(\frac{1}{2}\mathrm{x} + 3\right) - \left(\frac{2}{3}\mathrm{x} - 5\right)\)
• Need to: Simplify to match one of the answer choices

2. INFER the approach

• Key insight: The minus sign in front of the second expression must be distributed to BOTH terms inside the parentheses
• Strategy: Distribute first, then combine like terms

3. SIMPLIFY by distributing the minus sign

• \(\left(\frac{1}{2}\mathrm{x} + 3\right) - \left(\frac{2}{3}\mathrm{x} - 5\right)\)
• \(= \left(\frac{1}{2}\mathrm{x} + 3\right) - \frac{2}{3}\mathrm{x} + 5\)
• Note: The -5 becomes +5 when we distribute the negative

4. SIMPLIFY by rearranging and combining like terms

• \(= \frac{1}{2}\mathrm{x} - \frac{2}{3}\mathrm{x} + 3 + 5\)
• \(= \left(\frac{1}{2} - \frac{2}{3}\right)\mathrm{x} + 8\)

5. SIMPLIFY the fraction subtraction

• Find common denominator: \(\frac{1}{2} = \frac{3}{6}\) and \(\frac{2}{3} = \frac{4}{6}\)
• \(\frac{1}{2} - \frac{2}{3} = \frac{3}{6} - \frac{4}{6} = -\frac{1}{6}\)
• Final result: \(-\frac{1}{6}\mathrm{x} + 8\)

Answer: A. \(-\frac{1}{6}\mathrm{x} + 8\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students fail to distribute the minus sign to both terms in the second expression.

They incorrectly simplify \(\left(\frac{1}{2}\mathrm{x} + 3\right) - \left(\frac{2}{3}\mathrm{x} - 5\right)\) as:
\(\frac{1}{2}\mathrm{x} + 3 - \frac{2}{3}\mathrm{x} - 5 = -\frac{1}{6}\mathrm{x} - 2\)

This may lead them to select Choice B (\(-\frac{1}{6}\mathrm{x} - 2\))

Second Most Common Error:

Poor INFER reasoning: Students misinterpret the subtraction as multiplication.

Instead of subtracting the second expression, they multiply the two expressions together, leading to a quadratic result with x² terms.

This may lead them to select Choice C (\(-\frac{1}{3}\mathrm{x}^2 + \frac{1}{2}\mathrm{x} + 15\)) or Choice D (\(-\frac{1}{3}\mathrm{x}^2 - \frac{9}{2}\mathrm{x} - 15\))

The Bottom Line:

This problem tests whether students understand that subtracting an expression means distributing the negative sign to every term in that expression. The key challenge is remembering that \(-\left(\frac{2}{3}\mathrm{x} - 5\right)\) equals \(-\frac{2}{3}\mathrm{x} + 5\), not \(-\frac{2}{3}\mathrm{x} - 5\).