|-5x + 13| = 73 What is the sum of the solutions to the given equation?...

1. TRANSLATE the problem information

• Given equation: \(|-5\mathrm{x} + 13| = 73\)
• Find: Sum of all solutions

2. CONSIDER ALL CASES from absolute value definition

• The absolute value \(|\mathrm{A}| = \mathrm{B}\) means \(\mathrm{A} = \mathrm{B}\) or \(\mathrm{A} = -\mathrm{B}\)
• This creates two separate equations to solve:
  • Case 1: \(-5\mathrm{x} + 13 = 73\)
  • Case 2: \(-5\mathrm{x} + 13 = -73\)

3. SIMPLIFY Case 1: \(-5\mathrm{x} + 13 = 73\)

• Subtract 13 from both sides: \(-5\mathrm{x} = 60\)
• Divide both sides by -5: \(\mathrm{x} = -12\)

4. SIMPLIFY Case 2: \(-5\mathrm{x} + 13 = -73\)

• Subtract 13 from both sides: \(-5\mathrm{x} = -86\)
• Divide both sides by -5: \(\mathrm{x} = \frac{86}{5}\)

5. INFER what the question asks for

• We have two solutions: \(\mathrm{x} = -12\) and \(\mathrm{x} = \frac{86}{5}\)
• Question asks for sum of solutions, not individual solutions

6. SIMPLIFY the final calculation

• \(\mathrm{Sum} = -12 + \frac{86}{5}\)
• Convert to common denominator: \(-12 = -\frac{60}{5}\)
• \(\mathrm{Sum} = -\frac{60}{5} + \frac{86}{5} = \frac{26}{5}\)

Answer: D. \(\frac{26}{5}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students solve only one case from the absolute value equation, typically choosing \(-5\mathrm{x} + 13 = 73\) and finding \(\mathrm{x} = -12\). They miss that absolute value creates a second case where \(-5\mathrm{x} + 13 = -73\).

This incomplete solution may lead them to select Choice B (\(-12\)) since they only found one solution and didn't sum anything.

Second Most Common Error:

Poor INFER reasoning about the question: Students correctly find both solutions (\(-12\) and \(\frac{86}{5}\)) but misinterpret what the question asks for. They might select one of the individual solutions rather than computing their sum.

This may lead them to select Choice B (\(-12\)) or get confused trying to match \(\frac{86}{5}\) to the choices.

The Bottom Line:

Absolute value equations require systematic case analysis. Students must recognize that \(|\mathrm{A}| = \mathrm{B}\) always generates two equations to solve, then carefully read whether the question wants individual solutions or their sum.