Question:The equation |3x + 9| = 15 has two solutions. What is the sum of these two solutions?

1. TRANSLATE the absolute value equation

• Given: \(|3x + 9| = 15\)
• This means the expression \((3x + 9)\) has a distance of 15 from zero

2. CONSIDER ALL CASES for the absolute value

• Since \(|\mathrm{expression}| = 15\), we need two cases:
  • Case 1: The expression equals +15
  • Case 2: The expression equals -15
• This gives us:
  • \(3x + 9 = 15\)
  • \(3x + 9 = -15\)

3. SIMPLIFY Case 1: \(3x + 9 = 15\)

• Subtract 9 from both sides: \(3x = 6\)
• Divide by 3: \(x = 2\)

4. SIMPLIFY Case 2: \(3x + 9 = -15\)

• Subtract 9 from both sides: \(3x = -24\)
• Divide by 3: \(x = -8\)

5. Verify both solutions

• For \(x = 2\): \(|3(2) + 9| = |15| = 15\) ✓
• For \(x = -8\): \(|3(-8) + 9| = |-15| = 15\) ✓

6. Find the sum of solutions

• Sum = \(2 + (-8) = -6\)

Answer: -6

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students solve only the positive case \(3x + 9 = 15\), getting \(x = 2\), and forget there's a second case.

They might think "the absolute value equals 15, so \(3x + 9 = 15\)" without realizing that \(3x + 9\) could also equal -15 to make the absolute value equal 15. This leads them to find only one solution (\(x = 2\)) and either give that as their final answer or become confused about what to do next.

This leads to confusion and an incomplete solution.

Second Most Common Error:

Poor TRANSLATE reasoning: Students incorrectly set up the negative case as \(3x + 9 = -15\) but then solve it as \(3x = -15 + 9 = -6\), getting \(x = -2\).

This sign error occurs when students rush through the algebra without carefully tracking negative signs. When they check \(x = -2\), they get \(|3(-2) + 9| = |3| = 3 \neq 15\), which should alert them to their error, but they might not verify their solutions.

This may lead them to calculate an incorrect sum or abandon the problem entirely.

The Bottom Line:

The key insight is recognizing that absolute value equations create two separate cases to solve. Students who only consider one case miss half the solution, while those who make algebraic errors in either case will get incorrect solutions that don't verify.