How do you find the range of x in a triangle in geometry?

Ok, well let me start you off with an example first.

Suppose you have triangle ABC with side lengths 6 and 15. What is the range of the possible values of the third side?

What I would do first is sketch a picture of triangle ABC and assign 6 and 15 to any two sides, it doesn't matter what two sides and I'll show you why in a moment.

To solve this problem, you need to set up an inequality using the Triangle Inequality Theorem. This theorem simply states that any two sides of a triangle must add up to be greater than the third side. To do this, we need to set up two inequalities:

6 + 15 < x and 6 + x > 15.

Simplify both those inequalities to solve for x. You should get x>21 and x>9. Now, all you need to do is set up a compound inequality using those two inequalities you just simplified and you should get:

9<x<21

Easy right? :)