|7 - 2x| = 15 What is the sum of the solutions to the given equation?...

1. TRANSLATE the problem information

• Given equation: \(|7 - 2\mathrm{x}| = 15\)
• Need to find: Sum of all solutions

2. CONSIDER ALL CASES for the absolute value equation

• Since the absolute value equals a positive number (15), the expression inside can be either positive or negative
• This creates two separate equations to solve:
  • Case 1: \(7 - 2\mathrm{x} = 15\) (when the inside expression equals +15)
  • Case 2: \(7 - 2\mathrm{x} = -15\) (when the inside expression equals -15)

3. SIMPLIFY Case 1: \(7 - 2\mathrm{x} = 15\)

• Subtract 7 from both sides: \(-2\mathrm{x} = 8\)
• Divide by -2: \(\mathrm{x} = -4\)

4. SIMPLIFY Case 2: \(7 - 2\mathrm{x} = -15\)

• Subtract 7 from both sides: \(-2\mathrm{x} = -22\)
• Divide by -2: \(\mathrm{x} = 11\)

5. Verify both solutions

• For \(\mathrm{x} = -4\): \(|7 - 2(-4)| = |7 + 8| = |15| = 15\) ✓
• For \(\mathrm{x} = 11\): \(|7 - 2(11)| = |7 - 22| = |-15| = 15\) ✓

6. Calculate the sum of solutions

• Sum = \(-4 + 11 = 7\)

Answer: D (7)

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 \(7 - 2\mathrm{x} = 15\), getting \(\mathrm{x} = -4\). They don't recognize that absolute value equations require checking both positive and negative possibilities for the expression inside.

This leads them to think -4 is the only solution and select Choice B (-4), or they might only solve the second case and select Choice E (11).

Second Most Common Error:

Poor SIMPLIFY execution: Students set up both cases correctly but make algebraic errors when solving the linear equations, particularly with sign errors when dividing by -2.

This leads to incorrect individual solutions, causing confusion about which answer choice represents the sum, often resulting in guessing.

The Bottom Line:

The key insight is recognizing that absolute value equations create two scenarios - the expression inside can equal either the positive or negative value. Missing this fundamental property of absolute values means missing half the solution.