|x + 45| = 48 What is the positive solution to the given equation?...

1. TRANSLATE the problem information

• Given: \(|\mathrm{x} + 45| = 48\)
• Find: the positive solution

2. CONSIDER ALL CASES for the absolute value equation

• When we have \(|\mathrm{A}| = \mathrm{B}\), this means the expression A inside the absolute value bars can equal either +B or -B
• So \(|\mathrm{x} + 45| = 48\) gives us two cases:
  • Case 1: \(\mathrm{x} + 45 = 48\)
  • Case 2: \(\mathrm{x} + 45 = -48\)

3. SIMPLIFY each case separately

• Case 1: \(\mathrm{x} + 45 = 48\)
  Subtract 45 from both sides: \(\mathrm{x} = 48 - 45 = 3\)

• Case 2: \(\mathrm{x} + 45 = -48\)
  Subtract 45 from both sides: \(\mathrm{x} = -48 - 45 = -93\)

4. APPLY CONSTRAINTS to select the final answer

• We found two solutions: \(\mathrm{x} = 3\) and \(\mathrm{x} = -93\)
• The question asks for the positive solution
• Since \(3 \gt 0\) and \(-93 \lt 0\), the positive solution is \(\mathrm{x} = 3\)

Answer: A. 3

Why Students Usually Falter on This Problem

Most Common Error Path:

Missing CONSIDER ALL CASES skill: Students often only solve for the positive case, thinking \(|\mathrm{x} + 45| = 48\) simply means \(\mathrm{x} + 45 = 48\). They get \(\mathrm{x} = 3\) and stop there, not realizing they should also solve \(\mathrm{x} + 45 = -48\).

While this actually leads to the correct final answer in this problem, it represents incomplete understanding that would cause errors in other absolute value problems.

Second Most Common Error:

Weak SIMPLIFY execution: Students make arithmetic errors when subtracting, such as:

• For \(\mathrm{x} + 45 = 48\): forgetting to subtract 45 and thinking \(\mathrm{x} = 48\)
• For \(\mathrm{x} + 45 = -48\): calculating \(\mathrm{x} = -48 - 45\) incorrectly

This may lead them to select Choice B (48) if they make the first error, or get confused and guess if they make calculation errors.

The Bottom Line:

Absolute value equations require systematic case analysis. Students who rush through without considering both possibilities, or who make basic arithmetic errors under pressure, will struggle with this problem type.