If |4x - 4| = 112, what is the positive value of x - 1?

1. CONSIDER ALL CASES for the absolute value equation

• Given: \(|4\mathrm{x} - 4| = 112\)
• The absolute value equation \(|\mathrm{A}| = \mathrm{B}\) (where \(\mathrm{B} \gt 0\)) always creates two cases:
  • Case 1: \(\mathrm{A} = \mathrm{B}\)
  • Case 2: \(\mathrm{A} = -\mathrm{B}\)
• This means we need to solve:
  • \(4\mathrm{x} - 4 = 112\)
  • \(4\mathrm{x} - 4 = -112\)

2. SIMPLIFY each case separately

Case 1: \(4\mathrm{x} - 4 = 112\)

• Add 4 to both sides: \(4\mathrm{x} = 116\)
• Divide both sides by 4: \(\mathrm{x} = 29\)
• Calculate \(\mathrm{x} - 1\): \(29 - 1 = 28\)

Case 2: \(4\mathrm{x} - 4 = -112\)

• Add 4 to both sides: \(4\mathrm{x} = -108\)
• Divide both sides by 4: \(\mathrm{x} = -27\)
• Calculate \(\mathrm{x} - 1\): \(-27 - 1 = -28\)

3. APPLY CONSTRAINTS to select the final answer

• We have two values for \(\mathrm{x} - 1\): 28 and -28
• The question asks specifically for the positive value
• Therefore: \(\mathrm{x} - 1 = 28\)

Answer: 28

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students often solve only one case of the absolute value equation, typically choosing \(4\mathrm{x} - 4 = 112\) and ignoring the negative case.

They get \(\mathrm{x} = 29\), calculate \(\mathrm{x} - 1 = 28\), and think they're done. While they happen to get the correct final answer, they miss half the solution process and don't realize there are actually two values of x that satisfy the original equation.

This incomplete approach works here by luck, but would cause them to miss points on problems that require showing both solutions.

Second Most Common Error:

Poor APPLY CONSTRAINTS reasoning: Students correctly find both cases and get \(\mathrm{x} - 1 = 28\) and \(\mathrm{x} - 1 = -28\), but then get confused about what "positive value" means.

Some students might think the question is asking for the positive x value (which would be \(\mathrm{x} = 29\)), rather than the positive value of the expression \(\mathrm{x} - 1\). This leads to confusion about what the final answer should be.

The Bottom Line:

Absolute value equations inherently create multiple cases, and students need to systematically work through all possibilities before applying any final constraints. The key insight is recognizing that \(|\mathrm{A}| = \mathrm{B}\) always means "A is exactly B units away from zero" - which happens when \(\mathrm{A} = \mathrm{B}\) or \(\mathrm{A} = -\mathrm{B}\).