Question:If |x - 1| = |x + 3|, what is one possible value of x?

1. TRANSLATE the problem information

• Given equation: \(|\mathrm{x} - 1| = |\mathrm{x} + 3|\)
• Need to find: One possible value of x
• What this tells us: We have two absolute value expressions that are equal

2. INFER the solution strategy

• Key insight: Two absolute values are equal when either:
  • The expressions inside are equal, OR
  • The expressions inside are opposites of each other
• This means we need to solve two separate cases

3. CONSIDER ALL CASES systematically

Case 1: Expressions inside are equal
\(\mathrm{x} - 1 = \mathrm{x} + 3\)

Case 2: Expressions inside are opposites
\(\mathrm{x} - 1 = -(\mathrm{x} + 3)\)

4. SIMPLIFY Case 1

\(\mathrm{x} - 1 = \mathrm{x} + 3\)
Subtract x from both sides: \(-1 = 3\)
This is impossible! No solution from Case 1.

5. SIMPLIFY Case 2

\(\mathrm{x} - 1 = -(\mathrm{x} + 3)\)
\(\mathrm{x} - 1 = -\mathrm{x} - 3\)
Add x to both sides: \(2\mathrm{x} - 1 = -3\)
Add 1 to both sides: \(2\mathrm{x} = -2\)
Divide by 2: \(\mathrm{x} = -1\)

6. Verify the solution

For \(\mathrm{x} = -1\): \(|(-1) - 1| = |-2| = 2\) and \(|(-1) + 3| = |2| = 2\) ✓

Answer: -1

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that absolute value equations require case analysis. They might try to solve \(|\mathrm{x} - 1| = |\mathrm{x} + 3|\) by simply removing the absolute value bars and solving \(\mathrm{x} - 1 = \mathrm{x} + 3\), which leads to the impossible equation \(-1 = 3\). This leads to confusion and guessing, or they conclude there's no solution when actually \(\mathrm{x} = -1\) exists.

Second Most Common Error:

Incomplete CONSIDER ALL CASES execution: Students recognize they need cases but only consider one case (usually the simpler \(\mathrm{x} - 1 = \mathrm{x} + 3\)) and stop when it yields no solution. They miss the critical second case where \(\mathrm{x} - 1 = -(\mathrm{x} + 3)\), which contains the actual answer. This causes them to conclude incorrectly that there's no solution.

The Bottom Line:

Absolute value equations are fundamentally about analyzing multiple scenarios. The key insight is that \(|\mathrm{A}| = |\mathrm{B}|\) doesn't just mean \(\mathrm{A} = \mathrm{B}\) – it means the distances are equal, which happens when \(\mathrm{A} = \mathrm{B}\) OR when A and B are opposites (\(\mathrm{A} = -\mathrm{B}\)). Missing this deeper understanding of what absolute value equality means is what trips up most students.