For what value of x is the equation |2x + 6| = 4x true?

1. INFER the key constraint

• Given: \(|2\mathrm{x} + 6| = 4\mathrm{x}\)
• Key insight: Since absolute values are always non-negative, \(4\mathrm{x} \geq 0\)
• This means \(\mathrm{x} \geq 0\) must be true for any valid solution

2. CONSIDER ALL CASES for the absolute value

• The expression \(|2\mathrm{x} + 6|\) has two possible forms:
  • If \(2\mathrm{x} + 6 \geq 0\), then \(|2\mathrm{x} + 6| = 2\mathrm{x} + 6\)
  • If \(2\mathrm{x} + 6 \lt 0\), then \(|2\mathrm{x} + 6| = -(2\mathrm{x} + 6) = -2\mathrm{x} - 6\)

3. SIMPLIFY Case 1: When \(2\mathrm{x} + 6 \geq 0\)

• Set up equation: \(2\mathrm{x} + 6 = 4\mathrm{x}\)
• Subtract \(2\mathrm{x}\): \(6 = 2\mathrm{x}\)
• Divide by 2: \(\mathrm{x} = 3\)

4. APPLY CONSTRAINTS to check Case 1 validity

• Check case condition: Is \(2\mathrm{x} + 6 \geq 0\) when \(\mathrm{x} = 3\)?
  → \(2(3) + 6 = 12 \geq 0\) ✓
• Check original constraint: Is \(4\mathrm{x} \geq 0\) when \(\mathrm{x} = 3\)?
  → \(4(3) = 12 \geq 0\) ✓
• Verify in original equation: \(|2(3) + 6| = |12| = 12\) and \(4(3) = 12\) ✓

5. SIMPLIFY Case 2: When \(2\mathrm{x} + 6 \lt 0\)

• Set up equation: \(-2\mathrm{x} - 6 = 4\mathrm{x}\)
• Add \(2\mathrm{x}\): \(-6 = 6\mathrm{x}\)
• Divide by 6: \(\mathrm{x} = -1\)

6. APPLY CONSTRAINTS to check Case 2 validity

• Check case condition: Is \(2\mathrm{x} + 6 \lt 0\) when \(\mathrm{x} = -1\)?
  → \(2(-1) + 6 = 4\), which is NOT \(\lt 0\) ✗
• Check original constraint: Is \(4\mathrm{x} \geq 0\) when \(\mathrm{x} = -1\)?
  → \(4(-1) = -4 \lt 0\) ✗

Answer: 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students solve only one case, typically treating the absolute value equation like \(|2\mathrm{x} + 6| = 4\mathrm{x}\) becomes \(2\mathrm{x} + 6 = 4\mathrm{x}\) directly.

They get \(\mathrm{x} = 3\), verify it works, and stop there without considering the second case. While this leads to the correct answer in this specific problem, it shows incomplete understanding of absolute value equations and would fail on problems where the 'obvious' case doesn't yield the answer.

This approach demonstrates procedural luck rather than conceptual mastery.

Second Most Common Error:

Missing APPLY CONSTRAINTS reasoning: Students correctly set up both cases but fail to check whether solutions satisfy the necessary constraints (both the case conditions and \(4\mathrm{x} \geq 0\)).

For Case 2, they might accept \(\mathrm{x} = -1\) as valid without recognizing that it violates both the case condition (\(2\mathrm{x} + 6\) should be negative) and the fundamental constraint (\(4\mathrm{x}\) must be non-negative). This leads to incorrectly reporting multiple solutions or selecting \(\mathrm{x} = -1\) if it appears as an answer choice.

The Bottom Line:

This problem tests systematic thinking about absolute values. Success requires both complete case analysis and careful constraint checking—skills that distinguish procedural understanding from conceptual mastery.