The function g is defined by the equation \(\mathrm{g(x) = |3x - 84|}\). For which of the following values of...

1. TRANSLATE the problem information

• Given: \(\mathrm{g(x) = |3x - 84|}\) and we need \(\mathrm{g(k) = 4k}\)
• This means: \(\mathrm{|3k - 84| = 4k}\)

2. INFER the constraint requirement

• For absolute value equations \(\mathrm{|A| = B}\) to have solutions, we need \(\mathrm{B \geq 0}\)
• Since our equation is \(\mathrm{|3k - 84| = 4k}\), we must have \(\mathrm{4k \geq 0}\)
• Therefore: \(\mathrm{k \geq 0}\) (any negative k values will be extraneous)

3. CONSIDER ALL CASES for the absolute value

• The equation \(\mathrm{|3k - 84| = 4k}\) splits based on whether \(\mathrm{(3k - 84)}\) is positive or negative

Case 1: When \(\mathrm{3k - 84 \geq 0}\) (so \(\mathrm{k \geq 28}\))

• Then \(\mathrm{|3k - 84| = 3k - 84}\)
• Equation becomes: \(\mathrm{3k - 84 = 4k}\)
• SIMPLIFY: \(\mathrm{3k - 84 = 4k}\)
  \(\mathrm{-84 = k}\)
  \(\mathrm{k = -84}\)

Case 2: When \(\mathrm{3k - 84 \lt 0}\) (so \(\mathrm{k \lt 28}\))

• Then \(\mathrm{|3k - 84| = -(3k - 84) = 84 - 3k}\)
• Equation becomes: \(\mathrm{84 - 3k = 4k}\)
• SIMPLIFY: \(\mathrm{84 - 3k = 4k}\)
  \(\mathrm{84 = 7k}\)
  \(\mathrm{k = 12}\)

4. APPLY CONSTRAINTS to check validity

• From Case 1: \(\mathrm{k = -84}\)
  • Check: \(\mathrm{-84 \geq 0}\)? No ❌ (violates our constraint)
  • This is an extraneous solution

• From Case 2: \(\mathrm{k = 12}\)
  • Check: \(\mathrm{12 \geq 0}\)? Yes ✓ (satisfies our constraint)
  • Also check: \(\mathrm{12 \lt 28}\)? Yes ✓ (consistent with Case 2 condition)

5. SIMPLIFY by verification

• \(\mathrm{g(12) = |3(12) - 84|}\)
  \(\mathrm{= |36 - 84|}\)
  \(\mathrm{= |-48|}\)
  \(\mathrm{= 48}\)
• \(\mathrm{4k = 4(12) = 48}\) ✓

Answer: B. 12

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER reasoning: Students often forget that absolute value equations \(\mathrm{|A| = B}\) require \(\mathrm{B \geq 0}\) for solutions to exist. They jump straight into case analysis without establishing the constraint \(\mathrm{k \geq 0}\), then fail to recognize that negative solutions are extraneous.

Without this insight, they might accept \(\mathrm{k = -84}\) from Case 1 as a valid answer. Since -84 isn't among the choices, this leads to confusion and guessing, or they might incorrectly think the problem has no solution.

Second Most Common Error:

Inadequate CONSIDER ALL CASES execution: Students may only consider one case of the absolute value equation, typically the "positive case," leading them to find \(\mathrm{k = -84}\) and then get stuck when it doesn't match any answer choice.

This incomplete analysis causes them to abandon systematic solution and guess among the given options.

The Bottom Line:

This problem requires students to combine their knowledge of absolute value properties with systematic case analysis and constraint checking. The key insight is recognizing that the constraint \(\mathrm{k \geq 0}\) eliminates one of the algebraically valid solutions, making this more than just a routine absolute value equation.