Which expression is equivalent to \(\mathrm{15x + 2(3x - (x + 4))}\)? 17x - 8 18x - 8 19x -...

1. INFER the solution strategy

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

2. SIMPLIFY the innermost parentheses first

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

3. SIMPLIFY inside the remaining parentheses

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

4. SIMPLIFY by distributing the coefficient

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

5. SIMPLIFY by combining final like terms

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

Answer: (C) \(19x - 8\)

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

Following this error path:

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

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

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.