What value of x satisfies the equation |x + 3| = |x - 7|?

1. INFER the solution strategy

• Given: \(|\mathrm{x + 3}| = |\mathrm{x - 7}|\)
• Key insight: Absolute value expressions behave differently depending on whether the inside is positive or negative
• Strategy: Find critical points where expressions equal zero, then analyze each interval

2. CONSIDER ALL CASES by finding critical points

• Critical points occur when expressions inside equal zero:
  • \(\mathrm{x + 3 = 0}\) → \(\mathrm{x = -3}\)
  • \(\mathrm{x - 7 = 0}\) → \(\mathrm{x = 7}\)
• This creates three intervals to check: \(\mathrm{x \lt -3}\), \(\mathrm{-3 ≤ x \lt 7}\), and \(\mathrm{x ≥ 7}\)

3. CONSIDER ALL CASES for each interval

Case 1: \(\mathrm{x \lt -3}\)

• Both \(\mathrm{(x + 3)}\) and \(\mathrm{(x - 7)}\) are negative
• \(|\mathrm{x + 3}| = -(\mathrm{x + 3}) = -\mathrm{x} - 3\)
• \(|\mathrm{x - 7}| = -(\mathrm{x - 7}) = -\mathrm{x} + 7\)
• SIMPLIFY: \(-\mathrm{x} - 3 = -\mathrm{x} + 7\) → \(-3 = 7\) (impossible)

Case 2: \(\mathrm{-3 ≤ x \lt 7}\)

• \(\mathrm{(x + 3) ≥ 0}\) and \(\mathrm{(x - 7) \lt 0}\)
• \(|\mathrm{x + 3}| = \mathrm{x + 3}\)
• \(|\mathrm{x - 7}| = -(\mathrm{x - 7}) = -\mathrm{x} + 7\)
• SIMPLIFY: \(\mathrm{x + 3 = -x + 7}\) → \(\mathrm{2x = 4}\) → \(\mathrm{x = 2}\)
• Since \(\mathrm{-3 ≤ 2 \lt 7}\), this solution is valid

Case 3: \(\mathrm{x ≥ 7}\)

• Both \(\mathrm{(x + 3)}\) and \(\mathrm{(x - 7)}\) are non-negative
• \(|\mathrm{x + 3}| = \mathrm{x + 3}\)
• \(|\mathrm{x - 7}| = \mathrm{x - 7}\)
• SIMPLIFY: \(\mathrm{x + 3 = x - 7}\) → \(\mathrm{3 = -7}\) (impossible)

4. Verify the solution

• Check \(\mathrm{x = 2}\): \(|\mathrm{2 + 3}| = |5| = 5\) and \(|\mathrm{2 - 7}| = |-5| = 5\) ✓

Answer: 2

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students try to solve \(|\mathrm{x + 3}| = |\mathrm{x - 7}|\) by immediately setting \(\mathrm{x + 3 = x - 7}\), getting \(\mathrm{3 = -7}\) and concluding there's no solution.

They miss that absolute values can be equal in two ways: when the expressions are equal OR when they're opposites. Without systematic case analysis, they only check one possibility and get frustrated when it leads to an impossible equation.

This leads to confusion and guessing among the answer choices.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students correctly set up case analysis but make algebraic errors when solving \(\mathrm{2x = 4}\), potentially getting \(\mathrm{x = 1}\) or \(\mathrm{x = 4}\) instead of \(\mathrm{x = 2}\).

They may also incorrectly handle the signs when writing \(|\mathrm{x - 7}| = -(\mathrm{x - 7}) = -\mathrm{x} + 7\), writing it as \(-\mathrm{x} - 7\) instead.

This may lead them to select an incorrect numerical answer if available in the choices.

The Bottom Line:

This problem requires recognizing that absolute value equations need systematic case analysis rather than direct algebraic manipulation. The key insight is understanding how absolute value expressions change behavior at critical points.