|x - 23| = 7 What is the sum of the solutions to the given equation?...

1. INFER the mathematical meaning

• Given: \(|\mathrm{x} - 23| = 7\)
• Key insight: An absolute value equation \(|\mathrm{A}| = \mathrm{B}\) (where \(\mathrm{B} \gt 0\)) means the expression A can equal either \(+\mathrm{B}\) or \(-\mathrm{B}\)
• This creates two separate equations to solve

2. CONSIDER ALL CASES by setting up both equations

• Case 1: \(\mathrm{x} - 23 = 7\) (when the expression inside is positive)
• Case 2: \(\mathrm{x} - 23 = -7\) (when the expression inside is negative)

3. SIMPLIFY each case with basic algebra

• Case 1: \(\mathrm{x} - 23 = 7\) → \(\mathrm{x} = 30\)
• Case 2: \(\mathrm{x} - 23 = -7\) → \(\mathrm{x} = 16\)

4. SIMPLIFY to find the final answer

• Sum of solutions: \(30 + 16 = 46\)

Answer: 46

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES reasoning: Students solve only one case, typically \(\mathrm{x} - 23 = 7\), getting \(\mathrm{x} = 30\). They forget that absolute value creates two possibilities and stop after finding the first solution.

This leads them to answer 30 instead of recognizing they need to find both solutions and add them together.

Second Most Common Error:

Poor INFER skill about absolute value: Students may try to solve \(|\mathrm{x} - 23| = 7\) by "removing" the absolute value bars without understanding what this creates. Some attempt \(\mathrm{x} - 23 = 7\) and get confused, while others may incorrectly think \(\mathrm{x} = 7\) or \(\mathrm{x} = 23\).

This leads to confusion and guessing among the available answer choices.

The Bottom Line:

The key challenge is recognizing that absolute value equations naturally split into two cases. Students who miss this fundamental insight either solve incompletely or get lost in the problem structure entirely.