Question:\(\mathrm{g(x) = |36 - 4x|}\)The function g is defined by the given equation. For which of the following values of...

1. TRANSLATE the problem information

• Given: \(\mathrm{g(x) = |36 - 4x|}\)
• Need to find: Value of m where \(\mathrm{g(m) = 5m}\)
• This means: \(\mathrm{|36 - 4m| = 5m}\)

2. INFER a crucial constraint

• Since absolute values are always non-negative: \(\mathrm{|36 - 4m| \geq 0}\)
• Since \(\mathrm{|36 - 4m| = 5m}\), we need: \(\mathrm{5m \geq 0}\)
• Therefore: \(\mathrm{m \geq 0}\) (this will help us check our solutions later)

3. CONSIDER ALL CASES for the absolute value

The expression \(\mathrm{|36 - 4m|}\) behaves differently depending on whether what's inside is positive or negative.

Case 1: When \(\mathrm{36 - 4m \geq 0}\)

• This happens when \(\mathrm{m \leq 9}\)
• Then \(\mathrm{|36 - 4m| = 36 - 4m}\)
• Our equation becomes: \(\mathrm{36 - 4m = 5m}\)

4. SIMPLIFY Case 1

\(\mathrm{36 - 4m = 5m}\)

\(\mathrm{36 = 5m + 4m}\)

\(\mathrm{36 = 9m}\)

\(\mathrm{m = 4}\)

5. APPLY CONSTRAINTS to verify Case 1

• Check: Is \(\mathrm{4 \leq 9}\)? Yes ✓ (satisfies our case condition)
• Check: Is \(\mathrm{4 \geq 0}\)? Yes ✓ (satisfies our constraint)
• Double-check: \(\mathrm{g(4) = |36 - 16| = 20}\), and \(\mathrm{5(4) = 20}\) ✓

Case 2: When \(\mathrm{36 - 4m \lt 0}\)

• This happens when \(\mathrm{m \gt 9}\)
• Then \(\mathrm{|36 - 4m| = -(36 - 4m) = 4m - 36}\)
• Our equation becomes: \(\mathrm{4m - 36 = 5m}\)

6. SIMPLIFY Case 2

\(\mathrm{4m - 36 = 5m}\)

\(\mathrm{-36 = 5m - 4m}\)

\(\mathrm{-36 = m}\)

\(\mathrm{m = -36}\)

7. APPLY CONSTRAINTS to verify Case 2

• Check: Is \(\mathrm{-36 \gt 9}\)? No ✗ (doesn't satisfy case condition)
• Check: Is \(\mathrm{-36 \geq 0}\)? No ✗ (violates our constraint)
• This solution is invalid

Answer: (A) 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students often try to solve \(\mathrm{|36 - 4m| = 5m}\) by just removing the absolute value bars without considering that the expression inside might be negative. They write \(\mathrm{36 - 4m = 5m}\), get \(\mathrm{m = 4}\), and stop there.

While this happens to give the right answer in this case, it's incomplete reasoning that would fail on other problems. Some students might also try \(\mathrm{-36 + 4m = 5m}\) and get confused about which approach is correct.

This incomplete case analysis leads to either getting lucky with the right answer or confusion and guessing.

Second Most Common Error:

Poor INFER reasoning: Students might solve both cases correctly but miss the constraint that \(\mathrm{m \geq 0}\) (since \(\mathrm{5m}\) must be non-negative). They could include both \(\mathrm{m = 4}\) and \(\mathrm{m = -36}\) as valid solutions and get confused about which answer choice to pick.

This may lead them to select Choice (E) 36 if they try to find some other relationship, or causes them to get stuck and guess.

The Bottom Line:

This problem requires systematic case analysis combined with constraint checking. Students who rush through without considering all possibilities for the absolute value, or who solve correctly but don't verify their solutions against the problem's inherent constraints, will struggle to confidently select the right answer.