98 points math! find the range of possible values for the variable. show work

Let be the length of the leg with one tick mark and the length of the leg with two tick marks.

In the upper triangle, the law of cosines says

In the lower triangle, it says

Subtract the second equation from the first to eliminate :

and are lengths so they must both be positive. 10 is also positive, so in order to preserve the sign on both sides of this equation, we must have

Now we have to be a bit careful. If is an acute angle, then as gets larger, the value of gets smaller. So if we have two angles and , with , then we would have .

This means in our inequality, taking the inverse cosine of both sides would reverse the inequality:

We know that is an angle in a triangle, so it must be some positive measure:

So we must have