We are given three expressions for measures of the sides of a triangle.
x-2, 10, 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):& x-2+ 10> 12
(II):& x-2+ 12> 10
(III):& 10+ 12> x-2
Let's solve each inequality for x.
Inequality
Simplified
Solution Set
x-2+10 > 12
x > 4
x>4
x-2+12 > 10
x > 0
x>0
10+12 > x-2
24 > x
x<24
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 24. Therefore, the possible values of x are the following.
4
