We are given three expressions for measures of the sides of a triangle.
x+2, x+4, x+6
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+ x+4> x+6
(II):& x+2+ x+6> x+4
(III):& x+4+ x+6> x+2
Let's solve each inequality for x.
Inequality
Simplified
Solution Set
x+2+x+4 > x+6
x>0
x>0
x+2+x+6 > x+4
x > -4
x>-4
x+4+x+6 > x+2
x>-8
x>-8
Now we can graph the inequalities and find the set of common solutions.
As we can see, the lines overlap after 0. Therefore, the possible values of x are the following.
x>0
