The function g is defined by \(\mathrm{g(x) = |x - 2x|}\). What value of b satisfies \(\mathrm{g(b) - g(4) =...

1. SIMPLIFY the function definition

• Given: \(\mathrm{g(x) = |x - 2x|}\)
• Combine like terms inside the absolute value: \(\mathrm{x - 2x = -x}\)
• Apply absolute value property: \(\mathrm{|-x| = |x|}\)
• Simplified function: \(\mathrm{g(x) = |x|}\)

2. TRANSLATE the problem requirements

• Need to find value of b where: \(\mathrm{g(b) - g(4) = 8}\)
• First, I need \(\mathrm{g(4)}\) using our simplified function

3. Evaluate \(\mathrm{g(4)}\)

• \(\mathrm{g(4) = |4| = 4}\)

4. TRANSLATE the equation with known values

• \(\mathrm{g(b) - g(4) = 8}\) becomes:
• \(\mathrm{g(b) - 4 = 8}\)
• Therefore: \(\mathrm{g(b) = 12}\)

5. INFER what \(\mathrm{g(b) = 12}\) means

• Since \(\mathrm{g(b) = |b|}\), I need to solve: \(\mathrm{|b| = 12}\)
• Key insight: This absolute value equation has two solutions

6. Solve the absolute value equation

• \(\mathrm{|b| = 12}\) means: \(\mathrm{b = 12}\) or \(\mathrm{b = -12}\)

7. APPLY CONSTRAINTS from answer choices

• Looking at choices: (A) -16, (B) 4, (C) 8, (D) 12
• Only \(\mathrm{b = 12}\) appears among the options

Answer: D

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students work with the unsimplified function \(\mathrm{g(x) = |x - 2x|}\) throughout the problem, making calculations unnecessarily complex and error-prone. They might try to substitute values directly into \(\mathrm{|b - 2b| = 12}\), leading to \(\mathrm{|-b| = 12}\), but then get confused about whether this equals \(\mathrm{|b| = 12}\) or treat it as a different equation entirely.

This leads to confusion and either guessing or selecting an incorrect systematic approach.

Second Most Common Error:

Poor INFER reasoning: Students correctly simplify to \(\mathrm{g(x) = |x|}\) and find \(\mathrm{g(b) = 12}\), but forget that absolute value equations typically have two solutions. They might only consider \(\mathrm{b = 12}\) or only consider \(\mathrm{b = -12}\), potentially missing the complete solution set. However, since only \(\mathrm{b = 12}\) appears in the choices, this error might accidentally lead to the correct answer through incomplete reasoning.

The Bottom Line:

This problem tests whether students can efficiently simplify absolute value expressions and understand that \(\mathrm{|x| = |-x|}\). The key insight is recognizing that \(\mathrm{|x - 2x| = |-x| = |x|}\), which transforms a potentially complex problem into a straightforward absolute value equation.