What is the larger solution to the given equation?|3x - 6| + 2 = 2|3x - 6| - 4Enter your...

1. TRANSLATE the problem information

• Given equation: \(|3\mathrm{x} - 6| + 2 = 2|3\mathrm{x} - 6| - 4\)
• Find: The larger solution

2. SIMPLIFY to isolate the absolute value

• Move all absolute value terms to one side:
  \(|3\mathrm{x} - 6| + 2 = 2|3\mathrm{x} - 6| - 4\)
  \(|3\mathrm{x} - 6| - 2|3\mathrm{x} - 6| = -4 - 2\)
  \(-|3\mathrm{x} - 6| = -6\)
• Multiply both sides by -1:
  \(|3\mathrm{x} - 6| = 6\)

3. CONSIDER ALL CASES for the absolute value equation

• Since \(|3\mathrm{x} - 6| = 6\), we need to solve both:

Case 1: \(3\mathrm{x} - 6 = 6\)

• Add 6: \(3\mathrm{x} = 12\)
• Divide by 3: \(\mathrm{x} = 4\)

Case 2: \(3\mathrm{x} - 6 = -6\)

• Add 6: \(3\mathrm{x} = 0\)
• Divide by 3: \(\mathrm{x} = 0\)

4. APPLY CONSTRAINTS to select the final answer

• Both solutions are valid: \(\mathrm{x} = 0\) and \(\mathrm{x} = 4\)
• The question asks for the larger solution
• Since \(4 \gt 0\), the larger solution is 4

Answer: 4

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students struggle with the algebraic manipulation step, particularly combining like terms with absolute value expressions. They might incorrectly handle \(-|3\mathrm{x} - 6| = -6\), forgetting to change signs when multiplying by -1, or make errors when collecting the absolute value terms on one side.

This leads to confusion about what equation to solve next, causing them to abandon systematic solution and guess.

Second Most Common Error:

Inadequate CONSIDER ALL CASES execution: Students successfully isolate \(|3\mathrm{x} - 6| = 6\) but only solve one case of the absolute value equation. They might solve \(3\mathrm{x} - 6 = 6\) to get \(\mathrm{x} = 4\), but forget that absolute value equations typically have two solutions, missing \(\mathrm{x} = 0\) entirely.

This causes them to think 4 is the only solution rather than the larger of two solutions.

The Bottom Line:

This problem challenges students' algebraic manipulation skills with absolute values and their understanding that absolute value equations require considering multiple cases. The combination of these two process skills makes this more complex than a standard absolute value equation.