If \(5(\mathrm{x} + 4) = 4(\mathrm{x} + 4) + 29\), what is the value of x + 4?

1. TRANSLATE the problem information

• Given equation: \(5(\mathrm{x} + 4) = 4(\mathrm{x} + 4) + 29\)
• Find: The value of \(\mathrm{x} + 4\) (not just x!)

2. INFER the most efficient approach

• Key insight: Notice that \((\mathrm{x} + 4)\) appears as a complete unit on both sides
• Instead of expanding with distributive property, treat \((\mathrm{x} + 4)\) as a single "variable"
• This lets us solve directly for \(\mathrm{x} + 4\) without finding x first

3. SIMPLIFY by subtracting \(4(\mathrm{x} + 4)\) from both sides

• \(5(\mathrm{x} + 4) - 4(\mathrm{x} + 4) = 4(\mathrm{x} + 4) + 29 - 4(\mathrm{x} + 4)\)
• The \(4(\mathrm{x} + 4)\) terms cancel on the right side:
• \([5 - 4](\mathrm{x} + 4) = 29\)
• \(1(\mathrm{x} + 4) = 29\)
• \(\mathrm{x} + 4 = 29\)

Answer: C. 29

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students automatically expand everything using distributive property instead of recognizing the elegant unit approach.

They expand: \(5\mathrm{x} + 20 = 4\mathrm{x} + 16 + 29\), then simplify to \(5\mathrm{x} + 20 = 4\mathrm{x} + 45\), then solve: \(\mathrm{x} = 25\). But the question asks for \(\mathrm{x} + 4\), not x! Since they found \(\mathrm{x} = 25\), they might stop there and not realize they need \(\mathrm{x} + 4 = 25 + 4 = 29\).

This may lead them to select Choice B (25).

Second Most Common Error:

Poor SIMPLIFY execution: Students make arithmetic errors when combining like terms or when subtracting from both sides, especially with the signs and coefficients.

This leads to confusion and incorrect calculations, causing them to select wrong answer choices or guess.

The Bottom Line:

The key insight is recognizing that when the same expression appears multiple times, you can often treat it as a single unit. This problem rewards strategic thinking over mechanical expansion - sometimes the elegant approach is much simpler than the "standard" method.