Question:|2x - 5| + 12 = 31What is the sum of the solutions to the given equation?Format: Fill-in-the-blank

1. TRANSLATE the problem information

• Given equation: \(|2\mathrm{x} - 5| + 12 = 31\)
• Need to find: The sum of all solutions to this equation
• This tells us we'll need to find multiple solutions and add them together

2. SIMPLIFY to isolate the absolute value

• Subtract 12 from both sides: \(|2\mathrm{x} - 5| = 31 - 12 = 19\)
• Now we have the clean form \(|2\mathrm{x} - 5| = 19\)

3. CONSIDER ALL CASES for the absolute value equation

• When \(|\mathrm{expression}| = 19\), we need both cases:
  • Case 1: The expression inside equals positive 19
  • Case 2: The expression inside equals negative 19

4. SIMPLIFY each case separately

Case 1: \(2\mathrm{x} - 5 = 19\)

• Add 5: \(2\mathrm{x} = 24\)
• Divide by 2: \(\mathrm{x} = 12\)

Case 2: \(2\mathrm{x} - 5 = -19\)

• Add 5: \(2\mathrm{x} = -14\)
• Divide by 2: \(\mathrm{x} = -7\)

5. TRANSLATE the final requirement

• Solutions are \(\mathrm{x} = 12\) and \(\mathrm{x} = -7\)
• Sum = \(12 + (-7) = 5\)

Answer: 5

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 \(2\mathrm{x} - 5 = 19\), getting \(\mathrm{x} = 12\). They forget that absolute value equations generally have two solutions and submit 12 as their final answer instead of finding both solutions and adding them.

Second Most Common Error:

Poor TRANSLATE reasoning: Students find both solutions correctly (\(\mathrm{x} = 12\) and \(\mathrm{x} = -7\)) but misunderstand what "sum of the solutions" means. They might subtract instead of add, getting \(12 - (-7) = 19\), or make sign errors when adding positive and negative numbers, possibly getting -5 instead of 5.

The Bottom Line:

This problem tests whether students understand the fundamental nature of absolute value equations—that they split into two cases—and whether they can carefully execute multi-step algebraic procedures while keeping track of what the final question is asking for.