If \(9(4 - 3\mathrm{x}) + 2 = 8(4 - 3\mathrm{x}) + 18\), what is the value of 4 - 3x?

1. INFER the most efficient approach

• Given: \(9(4 - 3\mathrm{x}) + 2 = 8(4 - 3\mathrm{x}) + 18\)
• Find: The value of \(4 - 3\mathrm{x}\)

Key insight: Since we need the value of \((4 - 3\mathrm{x})\) itself, not x, we can treat this entire expression as a single unit. This avoids the need to distribute and solve for x separately.

2. SIMPLIFY by isolating the expression

• Subtract 2 from both sides:
  \(9(4 - 3\mathrm{x}) = 8(4 - 3\mathrm{x}) + 16\)

• Subtract \(8(4 - 3\mathrm{x})\) from both sides:
  \(9(4 - 3\mathrm{x}) - 8(4 - 3\mathrm{x}) = 16\)
  \(1(4 - 3\mathrm{x}) = 16\)
  \(4 - 3\mathrm{x} = 16\)

Answer: D. 16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Not recognizing that \((4 - 3\mathrm{x})\) can be treated as a single unit

Students often think they must distribute first: \(9(4 - 3\mathrm{x})\) becomes \(36 - 27\mathrm{x}\), and \(8(4 - 3\mathrm{x})\) becomes \(32 - 24\mathrm{x}\). This creates the equation \(36 - 27\mathrm{x} + 2 = 32 - 24\mathrm{x} + 18\), leading to unnecessary complexity. After solving this longer route, they get \(\mathrm{x} = -4\) and may mistakenly select Choice B (-4) instead of recognizing they need to find \(4 - 3\mathrm{x}\), not x.

Second Most Common Error:

Poor SIMPLIFY execution: Making arithmetic errors during the isolation process

Students correctly identify the approach but make calculation mistakes, such as:

• Incorrectly computing \(18 - 2 = 14\) instead of 16
• Incorrectly computing \(9 - 8 = 0\) instead of 1

These errors lead to wrong final values and may cause them to select Choice A (-16) or Choice C (4).

The Bottom Line:

The key insight is recognizing when you can work with expressions as single units rather than expanding them. This problem rewards strategic thinking over computational complexity.