\((3\mathrm{x} + 7) - (\mathrm{x} - 4)\) Which of the following is equivalent to the given expression?...

1. INFER the solution strategy

• Given: \((3\mathrm{x} + 7) - (\mathrm{x} - 4)\)
• Strategy: First distribute the negative sign, then combine like terms

2. SIMPLIFY by distributing the negative sign

• \((3\mathrm{x} + 7) - (\mathrm{x} - 4)\)
• The minus sign applies to both terms inside the parentheses
• \(= 3\mathrm{x} + 7 - \mathrm{x} + 4\)
• Key insight: \(-(\mathrm{x} - 4) = -\mathrm{x} + 4\)

3. SIMPLIFY by identifying and grouping like terms

• \(= 3\mathrm{x} + 7 - \mathrm{x} + 4\)
• Group the x terms and constant terms: \((3\mathrm{x} - \mathrm{x}) + (7 + 4)\)

4. SIMPLIFY by combining like terms

• \(= (3\mathrm{x} - \mathrm{x}) + (7 + 4)\)
• \(= 2\mathrm{x} + 11\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Not properly distributing the negative sign

Students often write: \((3\mathrm{x} + 7) - (\mathrm{x} - 4) = 3\mathrm{x} + 7 - \mathrm{x} - 4 = 2\mathrm{x} + 3\)

They forget that the negative sign must be distributed to both terms inside the parentheses, treating \(-(\mathrm{x} - 4)\) as \(-\mathrm{x} - 4\) instead of \(-\mathrm{x} + 4\).

This leads them to select Choice B (2x + 3)

Second Most Common Error:

Poor INFER reasoning: Misunderstanding which terms to subtract

Some students incorrectly add all terms: \((3\mathrm{x} + 7) - (\mathrm{x} - 4)\) becomes \(3\mathrm{x} + 7 + \mathrm{x} + 4 = 4\mathrm{x} + 11\)

They fail to recognize that the subtraction applies to the entire expression (x - 4).

This leads them to select Choice D (4x + 11)

The Bottom Line:

The key challenge is correctly handling the distributive property with subtraction. The negative sign must distribute to create -x + 4, not -x - 4.