The function r is defined by \(\mathrm{r(x) = |15 - 2x|}\). What is the value of \(\mathrm{r(11)}\)?

1. TRANSLATE the problem information

• Given: Function \(\mathrm{r(x) = |15 - 2x|}\)
• Need to find: \(\mathrm{r(11)}\)
• What this means: Substitute \(\mathrm{x = 11}\) into the function

2. TRANSLATE the substitution

• Replace every x with 11:
• \(\mathrm{r(11) = |15 - 2(11)|}\)

3. SIMPLIFY inside the absolute value first

• \(\mathrm{r(11) = |15 - 2(11)|}\)
• \(\mathrm{r(11) = |15 - 22|}\)
• \(\mathrm{r(11) = |-7|}\)

4. SIMPLIFY by applying absolute value

• The absolute value of -7 is 7
• \(\mathrm{r(11) = 7}\)

Answer: C) 7

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students incorrectly think that \(\mathrm{|-7| = -7}\), not understanding that absolute value always gives a non-negative result.

They might reason: "I got -7 inside the absolute value bars, so the answer is -7." This leads them to select Choice A (-7).

Second Most Common Error:

Poor TRANSLATE reasoning: Students might substitute incorrectly, perhaps computing \(\mathrm{r(11) = |15 - 11|}\) instead of \(\mathrm{r(11) = |15 - 2(11)|}\), missing the coefficient 2.

This gives \(\mathrm{|15 - 11| = |4| = 4}\), which isn't among the answer choices, leading to confusion and guessing.

The Bottom Line:

This problem tests whether students understand both function notation and absolute value properties. The key insight is that absolute value always produces a non-negative result, even when the expression inside evaluates to a negative number.