The equation 8 + 2|x - 5| = 33 - 3|x - 5| has two real solutions. What is the...

1. INFER the best approach

• Given equation: \(8 + 2|\mathrm{x} - 5| = 33 - 3|\mathrm{x} - 5|\)
• Key insight: Both sides contain the same absolute value expression \(|\mathrm{x} - 5|\)
• Strategy: Use substitution to replace \(|\mathrm{x} - 5|\) with a single variable to create a linear equation

2. TRANSLATE using substitution

• Let \(\mathrm{A} = |\mathrm{x} - 5|\)
• The equation becomes: \(8 + 2\mathrm{A} = 33 - 3\mathrm{A}\)
• This transforms our absolute value equation into a standard linear equation

3. SIMPLIFY the linear equation

• Add \(3\mathrm{A}\) to both sides: \(8 + 2\mathrm{A} + 3\mathrm{A} = 33\)
• Combine like terms: \(8 + 5\mathrm{A} = 33\)
• Subtract 8 from both sides: \(5\mathrm{A} = 25\)
• Divide by 5: \(\mathrm{A} = 5\)

4. TRANSLATE back to original variable

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

5. CONSIDER ALL CASES for the absolute value equation

• \(|\mathrm{x} - 5| = 5\) means the distance from \(\mathrm{x}\) to 5 equals 5
• This gives two cases:
  • Case 1: \(\mathrm{x} - 5 = 5\) → \(\mathrm{x} = 10\)
  • Case 2: \(\mathrm{x} - 5 = -5\) → \(\mathrm{x} = 0\)

6. APPLY CONSTRAINTS to select final answer

• Both solutions \(\mathrm{x} = 0\) and \(\mathrm{x} = 10\) are valid
• The question asks for the positive solution
• Therefore: \(\mathrm{x} = 10\)

Answer: D) 10

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students see the absolute value expressions and immediately try to split into cases without recognizing the substitution opportunity. They attempt to solve by considering all four combinations of \(|\mathrm{x} - 5|\) being positive or negative, leading to:

• Case 1: \(8 + 2(\mathrm{x} - 5) = 33 - 3(\mathrm{x} - 5)\)
• Case 2: \(8 + 2(\mathrm{x} - 5) = 33 - 3[-(\mathrm{x} - 5)]\)
• Case 3: \(8 + 2[-(\mathrm{x} - 5)] = 33 - 3(\mathrm{x} - 5)\)
• Case 4: \(8 + 2[-(\mathrm{x} - 5)] = 33 - 3[-(\mathrm{x} - 5)]\)

This creates unnecessary complexity and calculation errors, often leading to confusion and guessing.

Second Most Common Error:

Inadequate CONSIDER ALL CASES execution: Students correctly use substitution to find \(|\mathrm{x} - 5| = 5\) but then only solve one case (usually \(\mathrm{x} - 5 = 5\), giving \(\mathrm{x} = 10\)) and miss the second solution \(\mathrm{x} = 0\). Since both 0 and 10 appear in the answer choices, this incomplete work may lead them to select Choice D (10) as their final answer - which happens to be correct - but they miss understanding that there are actually two solutions.

The Bottom Line:

This problem rewards students who recognize patterns and use strategic substitution rather than brute-force case analysis. The key insight is seeing that the same absolute value expression appears on both sides of the equation.