If 3x - 27 = 24, what is the value of x - 9?

1. TRANSLATE the problem information

• Given: \(3\mathrm{x} - 27 = 24\)
• Find: The value of \(\mathrm{x} - 9\) (not x itself)

2. INFER the most efficient approach

• Instead of solving for x first, look at the structure of the equation
• Notice that \(3\mathrm{x} - 27\) can be written as \(3(\mathrm{x} - 9)\) using the distributive property
• This means we can find \(\mathrm{x} - 9\) directly!

3. SIMPLIFY by factoring and solving

• Rewrite the equation: \(3(\mathrm{x} - 9) = 24\)
• Divide both sides by 3:
  \(\mathrm{x} - 9 = \frac{24}{3} = 8\)

Answer: B. 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Not recognizing the factoring opportunity and instead solving for x first.

Students solve \(3\mathrm{x} - 27 = 24\) to get \(\mathrm{x} = 17\), then calculate \(\mathrm{x} - 9 = 17 - 9 = 8\). While this gives the correct answer, it's less efficient and creates more opportunities for arithmetic errors. Some students might solve correctly for \(\mathrm{x} = 17\) but then forget to subtract 9, potentially selecting Choice C (24) if they confuse their work.

Second Most Common Error:

Poor TRANSLATE understanding: Misreading what the question asks for.

Students might solve for x correctly (\(\mathrm{x} = 17\)) but then think this is the final answer, selecting Choice D (35) if they make calculation errors, or getting confused about which value to report.

The Bottom Line:

This problem rewards students who can see structural relationships in algebraic expressions. The key insight is recognizing that the question asks for exactly what appears (in factored form) on the left side of the equation.