Let x be a real number such that |2x + 11| = 5.Since absolute value equations of the form |expression|...

1. TRANSLATE the problem information

• Given information:
  • \(|2\mathrm{x} + 11| = 5\)
  • Need to find the sum of all possible values of x
• What this tells us: We're dealing with an absolute value equation set equal to a positive number

2. CONSIDER ALL CASES for the absolute value equation

• Key insight: When |expression| = positive number, the expression inside can equal either the positive or negative value
• This means we need to solve TWO equations:
  • Case 1: \(2\mathrm{x} + 11 = 5\)
  • Case 2: \(2\mathrm{x} + 11 = -5\)

3. SIMPLIFY Case 1

• Solve \(2\mathrm{x} + 11 = 5\):

\(2\mathrm{x} = 5 - 11\)

\(2\mathrm{x} = -6\)

\(\mathrm{x} = -3\)

4. SIMPLIFY Case 2

• Solve \(2\mathrm{x} + 11 = -5\):

\(2\mathrm{x} = -5 - 11\)

\(2\mathrm{x} = -16\)

\(\mathrm{x} = -8\)

5. TRANSLATE to find what's actually being asked

• The problem asks for the sum of all possible values
• Sum = \((-3) + (-8) = -11\)

Answer: -11

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak CONSIDER ALL CASES skill: Students solve only one case, typically \(2\mathrm{x} + 11 = 5\), getting \(\mathrm{x} = -3\). They forget that absolute value equations have two solutions and submit -3 as their final answer instead of finding both solutions and their sum.

Second Most Common Error:

Poor TRANSLATE reasoning: Students correctly find both solutions (\(\mathrm{x} = -3\) and \(\mathrm{x} = -8\)) but misread the question. Instead of finding the sum, they might try to provide both values separately or get confused about what the question is actually asking for, leading to incomplete or incorrect responses.

The Bottom Line:

This problem tests whether students truly understand that absolute value equations systematically produce two solutions, and whether they can follow through with the specific calculation requested (the sum) rather than just finding the individual solutions.