We are given three expressions for measures of the sides of a triangle.
8, x, 12
According to the Triangle Inequality Theorem, the sum of the lengths of any two sides of a triangle is always greater than the length of the third side. Using this theorem, we can write three true inequalities for this triangle.
(I):& 8+ x> 12
(II):& 8+ 12> x
(III):& x+ 12> 8
Let's solve each inequality for x.
Inequality
Simplified
Solution Set
8+x > 12
x > 4
x>4
8+12 > x
20 > x
x<20
x+12 > 8
x > -4
x>-4
Now we can graph the inequalities and find the set of common solutions.
As we can see, the lines overlap on the segment from 4 to 20. Therefore, the possible values of x are the following.
4
