|x - 2| = 9 What is one possible solution to the given equation?...

1. TRANSLATE the problem information

• Given: \(|\mathrm{x} - 2| = 9\)
• Find: One possible value of x

2. INFER the solution approach

• The absolute value \(|\mathrm{x} - 2| = 9\) means the expression \(\mathrm{x} - 2\) is exactly 9 units away from 0
• This can happen in two ways: \(\mathrm{x} - 2 = 9\) or \(\mathrm{x} - 2 = -9\)
• We need to CONSIDER ALL CASES by solving both equations

3. SIMPLIFY the first case

• If \(\mathrm{x} - 2 = 9\)
• Add 2 to both sides: \(\mathrm{x} = 9 + 2 = 11\)

4. SIMPLIFY the second case

• If \(\mathrm{x} - 2 = -9\)
• Add 2 to both sides: \(\mathrm{x} = -9 + 2 = -7\)

5. Verify both solutions work

• Check \(\mathrm{x} = 11\): \(|11 - 2| = |9| = 9\) ✓
• Check \(\mathrm{x} = -7\): \(|-7 - 2| = |-9| = 9\) ✓

Answer: 11 or -7 (either value is a correct response)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize that absolute value equations require splitting into two cases.

They might think "\(|\mathrm{x} - 2| = 9\)" means "\(\mathrm{x} - 2 = 9\)" and solve only \(\mathrm{x} - 2 = 9\) to get \(\mathrm{x} = 11\). They miss the crucial insight that absolute value represents distance, which can result from both positive and negative values inside the absolute value bars. This leads them to provide only one solution when two exist.

Second Most Common Error:

Poor CONSIDER ALL CASES execution: Students recognize they need two cases but make sign errors when setting them up.

They might write both cases as \(\mathrm{x} - 2 = 9\) and \(\mathrm{x} - 2 = 9\) (forgetting the negative), or incorrectly write \(\mathrm{x} - 2 = -9\) and \(\mathrm{x} + 2 = 9\) (changing the wrong sign). This leads to finding duplicate or incorrect solutions.

The Bottom Line:

Absolute value equations are fundamentally about recognizing that distance works both ways on the number line - a key conceptual leap that requires systematic case analysis rather than just algebraic manipulation.