Which expression is equivalent to \(15\mathrm{x} + 2(3\mathrm{x} - (\mathrm{x} + 4))\)?17x - 818x - 819x - 819x + 8

1. INFER the solution strategy

• Given: \(\mathrm{15x + 2(3x - (x + 4))}\)
• Key insight: Work from innermost parentheses outward
• This prevents errors and follows proper order of operations

2. SIMPLIFY the innermost parentheses first

• Focus on: \(\mathrm{-(x + 4)}\)
• Distribute the negative sign: \(\mathrm{-(x + 4) = -x - 4}\)
• Expression becomes: \(\mathrm{15x + 2(3x - x - 4)}\)

3. SIMPLIFY inside the remaining parentheses

• Combine like terms: \(\mathrm{3x - x - 4 = 2x - 4}\)
• Expression becomes: \(\mathrm{15x + 2(2x - 4)}\)

4. SIMPLIFY by distributing the coefficient

• Apply distributive property: \(\mathrm{2(2x - 4) = 4x - 8}\)
• Expression becomes: \(\mathrm{15x + 4x - 8}\)

5. SIMPLIFY by combining final like terms

• Combine x terms: \(\mathrm{15x + 4x = 19x}\)
• Final result: \(\mathrm{19x - 8}\)

Answer: (C) \(\mathrm{19x - 8}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Poor SIMPLIFY execution: Incorrectly distributing the negative sign in step 2

Students often treat \(\mathrm{-(x + 4)}\) as \(\mathrm{-x + 4}\) instead of \(\mathrm{-x - 4}\). This happens because they forget that the negative sign must be distributed to BOTH terms inside the parentheses.

Following this error path:

• \(\mathrm{15x + 2(3x - x + 4) = 15x + 2(2x + 4) = 15x + 4x + 8 = 19x + 8}\)

This may lead them to select Choice D (\(\mathrm{19x + 8}\))

Second Most Common Error:

Weak INFER reasoning: Not recognizing the need to work systematically from innermost parentheses

Students might try to distribute the 2 immediately or work left-to-right, creating confusion about which operations to perform first. This leads to disorganized work and computational errors.

This causes them to get stuck and randomly select an answer.

The Bottom Line:

The nested parentheses with a negative sign create a "double jeopardy" situation - students must both follow proper order of operations AND correctly distribute negative signs. Missing either skill typically leads to a systematic error that produces one of the wrong answer choices.