\(\mathrm{f(x) = |59 - 2x|}\) The function f is defined by the given equation. For which of the following values...

1. TRANSLATE the problem information

• Given information:
  • \(\mathrm{f(x) = |59 - 2x|}\)
  • Need to find k where \(\mathrm{f(k) = 3k}\)
• This means: \(\mathrm{|59 - 2k| = 3k}\)

2. INFER the approach for absolute value equations

• Key insight: When solving \(\mathrm{|A| = B}\), we need \(\mathrm{B \geq 0}\) first
• Since our equation is \(\mathrm{|59 - 2k| = 3k}\), we need \(\mathrm{3k \geq 0}\), so \(\mathrm{k \geq 0}\)
• Then we consider two cases: either the expression inside equals 3k, or its negative equals 3k

3. SIMPLIFY Case 1: \(\mathrm{59 - 2k = 3k}\)

• Add 2k to both sides: \(\mathrm{59 = 5k}\)
• Divide by 5: \(\mathrm{k = \frac{59}{5}}\)

4. SIMPLIFY Case 2: \(\mathrm{59 - 2k = -3k}\)

• Add 3k to both sides: \(\mathrm{59 + k = 0}\)
• Subtract 59: \(\mathrm{k = -59}\)

5. APPLY CONSTRAINTS to eliminate invalid solutions

• From our requirement \(\mathrm{k \geq 0}\), we must reject \(\mathrm{k = -59}\)
• Check \(\mathrm{k = \frac{59}{5}}\): \(\mathrm{f(\frac{59}{5}) = |59 - 2(\frac{59}{5})| = |\frac{177}{5}| = \frac{177}{5}}\)
• And \(\mathrm{3k = 3(\frac{59}{5}) = \frac{177}{5}}\) ✓

Answer: A. \(\mathrm{\frac{59}{5}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that absolute value equations require case analysis. They might try to "remove" the absolute value bars without considering both positive and negative cases.

They solve directly as: \(\mathrm{59 - 2k = 3k}\), getting \(\mathrm{k = \frac{59}{5}}\), but never verify this is the complete solution. While they get the right answer by luck, they miss the systematic approach and might fail on similar problems where both cases matter.

Second Most Common Error:

Poor APPLY CONSTRAINTS execution: Students correctly set up both cases but fail to check their solutions by substitution. They might accept \(\mathrm{k = -59}\) as valid without realizing it makes \(\mathrm{3k}\) negative, violating the absolute value equation structure.

This leads to confusion about which answer to select, often causing them to guess or select Choice D (59) by misremembering their work.

The Bottom Line:

This problem tests whether students understand that absolute value equations create multiple cases that must be systematically checked. Success requires both setting up the cases correctly and verifying which solutions actually work.