|x - 9| + 45 = 63 What is the sum of the solutions to the given equation?...

1. SIMPLIFY to isolate the absolute value

• Given: \(|\mathrm{x} - 9| + 45 = 63\)
• Subtract 45 from both sides: \(|\mathrm{x} - 9| = 18\)

2. INFER that absolute value creates two cases

• When we have \(|\mathrm{expression}| = \mathrm{positive\ number}\), this means:
  • expression = positive number, OR
  • expression = negative number
• So \(|\mathrm{x} - 9| = 18\) means: \(\mathrm{x} - 9 = 18\) OR \(\mathrm{x} - 9 = -18\)

3. CONSIDER ALL CASES by solving both equations

• First case: \(\mathrm{x} - 9 = 18\)
  • Add 9 to both sides: \(\mathrm{x} = 27\)
• Second case: \(\mathrm{x} - 9 = -18\)
  • Add 9 to both sides: \(\mathrm{x} = -9\)

4. SIMPLIFY to find the final answer

• Sum of solutions: \(27 + (-9) = 18\)

Answer: 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students solve \(|\mathrm{x} - 9| = 18\) by only considering \(\mathrm{x} - 9 = 18\), forgetting that absolute value equations typically have two solutions.

They find \(\mathrm{x} = 27\) as the only solution and give 27 as their final answer, not realizing they need to find both solutions and then add them together.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly identify both cases but make algebraic errors when solving the linear equations, particularly with the negative case \(\mathrm{x} - 9 = -18\).

Common mistake: \(\mathrm{x} - 9 = -18\) → \(\mathrm{x} = -18 - 9 = -27\) (instead of \(\mathrm{x} = -18 + 9 = -9\)). This leads to solutions 27 and -27, giving a sum of 0.

The Bottom Line:

This problem tests whether students understand that absolute value equations generally produce two solutions, and whether they can systematically work through both cases without computational errors.