The function g is defined by \(\mathrm{g(x) = |x^2 - 10|}\). There are two positive values of b that satisfy...

1. TRANSLATE the problem setup

• Given information:
  • Function: \(\mathrm{g(x) = |x² - 10|}\)
  • Equation: \(\mathrm{g(1) + g(b) = 15}\)
  • Need: the greater of two positive values of \(\mathrm{b}\)
• What this tells us: We need to find \(\mathrm{g(1)}\) first, then solve for \(\mathrm{g(b)}\)

2. Calculate g(1) by substitution

• \(\mathrm{g(1) = |1² - 10|}\)
• \(\mathrm{= |1 - 10|}\)
• \(\mathrm{= |-9|}\)
• \(\mathrm{= 9}\)

3. SIMPLIFY to find g(b)

• Substitute \(\mathrm{g(1) = 9}\) into the equation:
• \(\mathrm{9 + g(b) = 15}\)
• Solve: \(\mathrm{g(b) = 15 - 9 = 6}\)

4. TRANSLATE g(b) = 6 using the function definition

• We know \(\mathrm{g(b) = |b² - 10|}\) and \(\mathrm{g(b) = 6}\)
• So we must solve: \(\mathrm{|b² - 10| = 6}\)

5. CONSIDER ALL CASES for the absolute value equation

• For \(\mathrm{|A| = 6}\), we get \(\mathrm{A = 6}\) or \(\mathrm{A = -6}\)

Case 1: \(\mathrm{b² - 10 = 6}\)

• \(\mathrm{b² = 16}\)
• \(\mathrm{b = ±4}\)

Case 2: \(\mathrm{b² - 10 = -6}\)

• \(\mathrm{b² = 4}\)
• \(\mathrm{b = ±2}\)

6. APPLY CONSTRAINTS to select valid solutions

• From our four solutions \(\mathrm{\{4, -4, 2, -2\}}\), we need only positive values
• Positive solutions: \(\mathrm{b = 4}\) and \(\mathrm{b = 2}\)

7. INFER the final answer from the constraint

• The problem asks for "the greater of these two values"
• Between 4 and 2, the greater value is 4

Answer: C (4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students solve only one case of the absolute value equation, typically just \(\mathrm{b² - 10 = 6}\), getting \(\mathrm{b = ±4}\). They miss the second case \(\mathrm{b² - 10 = -6}\) entirely.

Without finding \(\mathrm{b = ±2}\), they incorrectly think there's only one positive solution (\(\mathrm{b = 4}\)) rather than two positive solutions. This creates confusion about what the problem is asking for, and they may second-guess whether 4 is actually correct or randomly select another answer.

Second Most Common Error:

Poor APPLY CONSTRAINTS execution: Students find all four solutions \(\mathrm{\{4, -4, 2, -2\}}\) but fail to consistently apply the "positive values" constraint. They might include negative solutions in their final consideration or forget which solutions were positive.

This may lead them to select Choice D (6) by incorrectly thinking one of the solutions was 6, or causes confusion about which values to compare.

The Bottom Line:

Absolute value equations naturally create multiple cases, and students must systematically work through both possibilities while carefully tracking which solutions meet the problem's constraints. The key insight is recognizing that "two positive values" confirms you should find exactly four total solutions before filtering.