Question:|w + 5| - 8 = 11What is the product of the solutions for w in the given equation?

1. SIMPLIFY to isolate the absolute value

• Given: \(|\mathrm{w} + 5| - 8 = 11\)
• Add 8 to both sides: \(|\mathrm{w} + 5| = 19\)
• Now we have a clean absolute value equation

2. CONSIDER ALL CASES for the absolute value equation

• When \(|\mathrm{w} + 5| = 19\), we get two scenarios:
  • Case 1: The expression inside is positive: \(\mathrm{w} + 5 = 19\)
  • Case 2: The expression inside is negative: \(\mathrm{w} + 5 = -19\)
• This is because absolute value measures distance from zero, so both \(+19\) and \(-19\) are 19 units away from zero

3. SIMPLIFY each case separately

• Case 1: \(\mathrm{w} + 5 = 19\)
  Subtract 5: \(\mathrm{w} = 14\)

• Case 2: \(\mathrm{w} + 5 = -19\)
  Subtract 5: \(\mathrm{w} = -24\)

4. INFER what the question is asking for

• The question asks for "the product of the solutions"
• We found two solutions: \(\mathrm{w} = 14\) and \(\mathrm{w} = -24\)
• Product = \(14 \times (-24) = -336\)

Answer: \(-336\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students solve \(|\mathrm{w} + 5| = 19\) by only considering the positive case: \(\mathrm{w} + 5 = 19\), getting \(\mathrm{w} = 14\). They miss the negative case entirely and either submit 14 as their final answer or get confused about what to do next. This leads to incomplete solution and likely guessing.

Second Most Common Error:

Poor INFER reasoning about the final step: Students correctly find both solutions (\(\mathrm{w} = 14\) and \(\mathrm{w} = -24\)) but misread the question. They might submit one of the individual solutions instead of calculating the product, or they might add the solutions instead of multiplying them.

The Bottom Line:

Absolute value equations inherently have two solutions when the right side is positive. The key insight is recognizing this dual nature and then carefully executing what the question asks for with those solutions.