Question:If |12 - 3x| = 42, what is the positive value of 6 - x?

1. TRANSLATE the problem information

• Given: \(|12 - 3\mathrm{x}| = 42\)
• Find: The positive value of \(6 - \mathrm{x}\)

2. CONSIDER ALL CASES for the absolute value equation

• Since \(|\mathrm{A}| = \mathrm{B}\) means \(\mathrm{A} = \mathrm{B}\) or \(\mathrm{A} = -\mathrm{B}\), we have two cases:
• Case 1: \(12 - 3\mathrm{x} = 42\)
• Case 2: \(12 - 3\mathrm{x} = -42\)

3. SIMPLIFY Case 1

• \(12 - 3\mathrm{x} = 42\)
• \(-3\mathrm{x} = 30\)
• \(\mathrm{x} = -10\)

4. SIMPLIFY Case 2

• \(12 - 3\mathrm{x} = -42\)
• \(-3\mathrm{x} = -54\)
• \(\mathrm{x} = 18\)

5. Calculate \(6 - \mathrm{x}\) for both solutions

• When \(\mathrm{x} = -10\): \(6 - \mathrm{x} = 6 - (-10) = 16\)
• When \(\mathrm{x} = 18\): \(6 - \mathrm{x} = 6 - 18 = -12\)

6. APPLY CONSTRAINTS to select the final answer

• The question asks for the positive value
• Between 16 and -12, the positive value is 16

Answer: 16

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students often forget that absolute value equations have two cases and only solve one case, typically \(12 - 3\mathrm{x} = 42\). They get \(\mathrm{x} = -10\), calculate \(6 - \mathrm{x} = 16\), and stop there. While they happen to get the right answer in this case, they miss the complete solution process and would fail on problems where the positive answer comes from the second case.

Second Most Common Error:

Poor SIMPLIFY execution: Students make sign errors when solving the equations, especially in Case 2 where they might incorrectly solve \(12 - 3\mathrm{x} = -42\) as \(-3\mathrm{x} = -42 + 12 = -30\), leading to \(\mathrm{x} = 10\). This gives \(6 - \mathrm{x} = 6 - 10 = -4\), and since they need a positive value, they might incorrectly take the absolute value to get 4, leading to confusion and guessing.

The Bottom Line:

Absolute value equations require systematic consideration of both cases. Students who skip cases or make algebraic errors will either get incomplete solutions or be forced to guess when their single answer doesn't make sense.