The triangle inequality theorem states that the sum of any two sides of a triangle must be greater than the...

1. TRANSLATE the triangle inequality theorem to our specific triangle

• Given information:
  • Triangle has sides of length 6 and 12
  • Third side has length x
  • Triangle inequality: sum of any two sides > third side

• This means we need three inequalities:
  • \(\mathrm{6 + 12 \gt x}\)
  • \(\mathrm{6 + x \gt 12}\)
  • \(\mathrm{12 + x \gt 6}\)

2. SIMPLIFY each inequality to find constraints on x

• First inequality: \(\mathrm{6 + 12 \gt x}\)
  • \(\mathrm{18 \gt x}\), or \(\mathrm{x \lt 18}\)

• Second inequality: \(\mathrm{6 + x \gt 12}\)
  • \(\mathrm{x \gt 12 - 6}\), or \(\mathrm{x \gt 6}\)

• Third inequality: \(\mathrm{12 + x \gt 6}\)
  • \(\mathrm{x \gt 6 - 12}\), or \(\mathrm{x \gt -6}\)

3. INFER which constraints are meaningful

• We have: \(\mathrm{x \lt 18}\), \(\mathrm{x \gt 6}\), and \(\mathrm{x \gt -6}\)
• Since side lengths are positive and \(\mathrm{x \gt 6}\) already requires x to be greater than -6, the constraint \(\mathrm{x \gt -6}\) doesn't add new information
• The meaningful constraints are: \(\mathrm{x \gt 6}\) AND \(\mathrm{x \lt 18}\)

4. TRANSLATE back to compound inequality notation

• \(\mathrm{x \gt 6}\) and \(\mathrm{x \lt 18}\) can be written as: \(\mathrm{6 \lt x \lt 18}\)

Answer: C. \(\mathrm{6 \lt x \lt 18}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often apply the triangle inequality incompletely, using only one or two of the three required inequalities instead of all three.

For instance, they might only check that \(\mathrm{6 + 12 \gt x}\), getting \(\mathrm{x \lt 18}\), and forget to check the other two conditions. This leads them to think that any value less than 18 would work, causing them to select Choice A (\(\mathrm{x \lt 18}\)).

Second Most Common Error:

Poor INFER reasoning: Students correctly set up all three inequalities but fail to combine the constraints properly, missing that they need both \(\mathrm{x \gt 6}\) AND \(\mathrm{x \lt 18}\) simultaneously.

They might think the constraints are alternatives rather than requirements that must all be satisfied together. This confusion about how to combine multiple constraints leads to guessing or selecting Choice D (\(\mathrm{x \lt 6}\) or \(\mathrm{x \gt 18}\)).

The Bottom Line:

The triangle inequality theorem requires checking ALL three possible combinations of sides, not just the most obvious one. Students must systematically apply the theorem and then correctly combine the resulting constraints.