|x - 5| = 10 What is one possible solution to the given equation?...

1. INFER what the absolute value equation means

• Given: \(|\mathrm{x} - 5| = 10\)
• By definition of absolute value, this means the expression inside the absolute value bars can equal either +10 or -10
• This gives us two separate equations to solve:
  • \(\mathrm{x} - 5 = 10\) (positive case)
  • \(\mathrm{x} - 5 = -10\) (negative case)

2. SIMPLIFY the first equation

• Start with: \(\mathrm{x} - 5 = 10\)
• Add 5 to both sides: \(\mathrm{x} = 15\)

3. SIMPLIFY the second equation

• Start with: \(\mathrm{x} - 5 = -10\)
• Add 5 to both sides: \(\mathrm{x} = -5\)

4. CONSIDER ALL CASES for the final answer

• Both solutions are valid since there are no restrictions
• The equation has two solutions: \(\mathrm{x} = 15\) and \(\mathrm{x} = -5\)

Answer: 15 or -5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students solve only the positive case (\(\mathrm{x} - 5 = 10\)) and get \(\mathrm{x} = 15\), but completely forget about the negative case.

They think "absolute value of something equals 10, so that something must equal 10" without realizing that if \(|\mathrm{a}| = 10\), then a could be either +10 or -10. This leads them to give only one answer (15) instead of recognizing both solutions.

Second Most Common Error:

Conceptual confusion about absolute value definition: Students might incorrectly think \(|\mathrm{x} - 5| = 10\) means \(\mathrm{x} - 5 = ±10\), then solve \(\mathrm{x} - 5 = ±10\) as a single equation, leading to algebraic confusion.

Instead of cleanly separating into two distinct cases, they get tangled up trying to work with the ± symbol, often making sign errors or getting confused about what operations to perform.

The Bottom Line:

Absolute value equations require systematic case analysis. The key insight is that |expression| = positive number always creates exactly two scenarios to check, and both solutions are typically valid unless the problem context introduces restrictions.