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).& AB+AC>CB
(II).& AB+CB>AC
(III).& AC+CB>AB
We are given that AB is 2x+22, AC is x+5, and CB is 5x-7. Let's substitute these values into the above inequalities and solve them for x.
Inequality
Substitution
Solution Set
AB+AC>CB
2x+22+x+5>5x-7
x<17
AB+CB>AC
2x+22+5x-7>x+5
x>- 5/3
AC+CB>AB
x+5+5x-7>2x+22
x>6
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 the values 6 to 17. With that said, the possible values of x can be written in the following inequality.
6
