The function f is defined by \(\mathrm{f(x) = |x - 4x|}\). What value of a satisfies \(\mathrm{f(5) - f(a) =...

1. SIMPLIFY the function definition

• Given: \(\mathrm{f(x) = |x - 4x|}\)
• Combine like terms inside absolute value: \(\mathrm{f(x) = |-3x|}\)
• Use absolute value property: \(\mathrm{f(x) = 3|x|}\)

2. TRANSLATE the given information

• We need \(\mathrm{f(5) - f(a) = -15}\)
• First calculate f(5): \(\mathrm{f(5) = 3|5| = 3(5) = 15}\)

3. INFER the value of f(a)

• Substitute into the equation: \(\mathrm{15 - f(a) = -15}\)
• Solve for f(a): \(\mathrm{f(a) = 15 - (-15) = 30}\)

4. SIMPLIFY to find the value of a

• Since \(\mathrm{f(a) = 3|a|}\), we have: \(\mathrm{3|a| = 30}\)
• Divide both sides by 3: \(\mathrm{|a| = 10}\)

5. CONSIDER ALL CASES for absolute value equation

• If \(\mathrm{|a| = 10}\), then \(\mathrm{a = 10}\) or \(\mathrm{a = -10}\)
• Check answer choices: Only \(\mathrm{a = 10}\) appears (Choice C)

Answer: C. 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students don't properly simplify \(\mathrm{|x - 4x|}\) and instead try to work with the more complex expression throughout the problem.

They might calculate f(5) as \(\mathrm{|5 - 4(5)| = |5 - 20| = |-15| = 15}\) (getting the right answer by luck), but then struggle with \(\mathrm{f(a) = |a - 4a| = |-3a|}\), not recognizing this equals \(\mathrm{3|a|}\). This leads to unnecessarily complicated algebra and potential calculation errors.

This may lead them to select Choice A (-20) after making sign errors, or abandon the systematic approach and guess.

Second Most Common Error:

Poor algebraic manipulation: Students make a sign error when solving \(\mathrm{f(5) - f(a) = -15}\), getting \(\mathrm{f(a) = 0}\) instead of \(\mathrm{f(a) = 30}\).

If \(\mathrm{f(a) = 0}\), then \(\mathrm{3|a| = 0}\), so \(\mathrm{|a| = 0}\), which means \(\mathrm{a = 0}\). Since 0 isn't among the choices, they may guess or select Choice B (5) thinking it's "close to zero."

The Bottom Line:

This problem tests whether students can efficiently simplify absolute value expressions and systematically work through multi-step equations. The key insight is recognizing that \(\mathrm{|x - 4x| = 3|x|}\) early in the solution.