\((3\mathrm{k} - 6) - (2\mathrm{k} + 1)\)Which of the following is equivalent to the expression above?

1. SIMPLIFY by distributing the negative sign

• Given: \((3\mathrm{k} - 6) - (2\mathrm{k} + 1)\)
• The key insight: When you subtract a parenthetical expression, you must distribute the negative sign to every term inside those parentheses
• SIMPLIFY the distribution: \((3\mathrm{k} - 6) - (2\mathrm{k} + 1) = 3\mathrm{k} - 6 - 2\mathrm{k} - 1\)

2. SIMPLIFY by grouping like terms

• Identify like terms: \(3\mathrm{k}\) and \(-2\mathrm{k}\) are like terms (both have variable k), \(-6\) and \(-1\) are like terms (both constants)
• Group them: \((3\mathrm{k} - 2\mathrm{k}) + (-6 - 1)\)

3. SIMPLIFY by combining like terms

• Combine the k-terms: \(3\mathrm{k} - 2\mathrm{k} = \mathrm{k}\)
• Combine the constants: \(-6 - 1 = -7\)
• Final result: \(\mathrm{k} - 7\)

Answer: C) k - 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly distribute the negative sign, treating \(-(2\mathrm{k} + 1)\) as \(-2\mathrm{k} + 1\) instead of \(-2\mathrm{k} - 1\).

When they write \((3\mathrm{k} - 6) - (2\mathrm{k} + 1) = 3\mathrm{k} - 6 - 2\mathrm{k} + 1\), they get:

• k-terms: \(3\mathrm{k} - 2\mathrm{k} = \mathrm{k}\) ✓
• Constants: \(-6 + 1 = -5\) ✗

This gives them \(\mathrm{k} - 5\), leading them to select Choice B (k - 5).

Second Most Common Error:

Poor SIMPLIFY reasoning: Students correctly distribute the negative sign but make arithmetic errors when combining constants.

They correctly get \(3\mathrm{k} - 6 - 2\mathrm{k} - 1\), but then miscalculate \(-6 - 1\) as \(-5\) instead of \(-7\), again leading to \(\mathrm{k} - 5\) and Choice B (k - 5).

The Bottom Line:

The negative sign distribution is the critical step that trips up most students. Remember: subtracting a binomial means subtracting both terms in that binomial, which requires distributing the negative sign to each term inside the parentheses.