The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
Using this theorem, we can write three inequalities that are true for this triangle.
(I):& XZ+ZY>YX
(II):& XZ+YX>ZY
(III):& YX+ZY>XZ
We are given that XZ is x+13, ZY is 2x+7, and YX is 4x-1. Let's substitute these values into the above inequalities and solve them for x.
Inequality
Substitution
Solution Set
XZ+ZY>YX
x+13+2x+7 > 4x-1
x<21
XZ+YX>ZY
x+13+4x-1>2x+7
x>- 5/3
YX+ZY>XZ
4x-1+2x+7>x+13
x>7/5
Using the three solution sets for x, let's find the common solutions for these three inequalities by graphing them on a number line.
As we can see, all three lines overlap on the segment from 75 to 21. Therefore, the possible values of x are as follows.
7/5
