Question:5|3 - x| = 35 - 2|3 - x|What is the positive solution to the given equation?-8-2358

1. INFER the solution strategy

• Given: \(5|3 - \mathrm{x}| = 35 - 2|3 - \mathrm{x}|\)
• Key insight: Both sides contain the same absolute value expression \(|3 - \mathrm{x}|\)
• Strategy: Use substitution to simplify before solving

2. SIMPLIFY using substitution

• Let \(\mathrm{u} = |3 - \mathrm{x}|\)
• The equation becomes: \(5\mathrm{u} = 35 - 2\mathrm{u}\)
• Add \(2\mathrm{u}\) to both sides: \(7\mathrm{u} = 35\)
• Divide by 7: \(\mathrm{u} = 5\)

3. TRANSLATE back to the original variable

• Since \(\mathrm{u} = |3 - \mathrm{x}|\), we have: \(|3 - \mathrm{x}| = 5\)

4. CONSIDER ALL CASES for the absolute value equation

• When \(|3 - \mathrm{x}| = 5\), we get two cases:
  • Case 1: \(3 - \mathrm{x} = 5\)
  • Case 2: \(3 - \mathrm{x} = -5\)

5. SIMPLIFY each case

• Case 1: \(3 - \mathrm{x} = 5\)
  • Subtract 3: \(-\mathrm{x} = 2\)
  • Multiply by -1: \(\mathrm{x} = -2\)

• Case 2: \(3 - \mathrm{x} = -5\)
  • Subtract 3: \(-\mathrm{x} = -8\)
  • Multiply by -1: \(\mathrm{x} = 8\)

6. APPLY CONSTRAINTS to select the final answer

• Both solutions work in the original equation
• The question asks for the positive solution
• Between \(\mathrm{x} = -2\) and \(\mathrm{x} = 8\), only \(\mathrm{x} = 8\) is positive

Answer: E) 8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students attempt to expand the equation directly without recognizing the substitution opportunity.

They might try to work with cases immediately: "If \(3 - \mathrm{x} \geq 0\), then..." and "If \(3 - \mathrm{x} \lt 0\), then..." This creates a complex system with multiple subcases and lengthy algebra, leading to confusion about which expressions to use on each side. This complexity causes them to make algebraic errors or abandon the systematic approach, leading to confusion and guessing.

Second Most Common Error:

Inadequate CONSIDER ALL CASES execution: Students solve for one case of \(|3 - \mathrm{x}| = 5\) but forget the second case.

They might solve \(3 - \mathrm{x} = 5\) to get \(\mathrm{x} = -2\), then stop without considering \(3 - \mathrm{x} = -5\). Since -2 is negative and they need the positive solution, this incomplete approach leaves them without a valid answer. This may lead them to select Choice B (-2) or causes them to get stuck and guess.

The Bottom Line:

The substitution insight makes this problem much simpler than it initially appears, but students who miss this strategy face unnecessarily complex algebra and are prone to errors in case analysis.