Given 22 and 50 as the lengths of two sides of a triangle, what is the range of possible values for the third side x?

What you are checking

To make a real triangle, the three side lengths must satisfy the triangle inequality, meaning any one side is shorter than the sum of the other two and longer than their difference.

Apply the triangle inequality to 22, 50, and $x$

With sides $22$, $50$, and $x$, the triangle inequality for $x$ is:

$$|50-22|<x<50+22$$

Compute the endpoints:

$$|50-22|=28 \quad \text{and} \quad 50+22=72$$

So the allowed range is:

$$28<x<72$$

Why the endpoints are not included

If $x=28$, the sides would line up in a straight line (degenerate triangle). If $x=72$, the same thing happens. A triangle needs positive area, so the inequalities are strict.