|2x - 1| = 7 What is the positive solution to the given equation?...

1. INFER the nature of absolute value equations

• Given: \(|2\mathrm{x} - 1| = 7\)
• Key insight: When \(|\mathrm{A}| = \mathrm{B}\) (where \(\mathrm{B} \gt 0\)), this means \(\mathrm{A} = \mathrm{B}\) OR \(\mathrm{A} = -\mathrm{B}\)
• This equation will have two possible solutions

2. CONSIDER ALL CASES for the absolute value

• Case 1: The expression inside the absolute value equals the positive value
  - \(2\mathrm{x} - 1 = 7\)
• Case 2: The expression inside the absolute value equals the negative value
  - \(2\mathrm{x} - 1 = -7\)

3. SIMPLIFY each case through algebraic steps

For Case 1: \(2\mathrm{x} - 1 = 7\)

• Add 1 to both sides: \(2\mathrm{x} = 8\)
• Divide by 2: \(\mathrm{x} = 4\)

For Case 2: \(2\mathrm{x} - 1 = -7\)

• Add 1 to both sides: \(2\mathrm{x} = -6\)
• Divide by 2: \(\mathrm{x} = -3\)

4. APPLY CONSTRAINTS to select the correct answer

• We found two solutions: \(\mathrm{x} = 4\) and \(\mathrm{x} = -3\)
• The question specifically asks for the "positive solution"
• Since \(4 \gt 0\) and \(-3 \lt 0\), the positive solution is \(\mathrm{x} = 4\)

Answer: C (4)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students solve only one case of the absolute value equation, typically just \(2\mathrm{x} - 1 = 7\), getting \(\mathrm{x} = 4\). While this happens to give the correct final answer, they miss the complete mathematical understanding that absolute value equations generally have two solutions.

Second Most Common Error:

Poor INFER reasoning: Students may not recognize that absolute value equations require considering both positive and negative cases. Instead, they might try to "remove" the absolute value bars without proper justification, leading to confusion about how to proceed systematically.

The Bottom Line:

Absolute value equations are fundamentally about understanding that \(|\mathrm{A}| = \mathrm{B}\) means "A is B units away from zero," which can happen in two directions. Success requires systematically considering both possibilities and then applying any given constraints.