We are given three expressions for measures of the sides of a triangle.
x, 4, 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+ 4> 6
(II):& x+ 6> 4
(III):& 4+ 6> x
Let's solve each inequality for x.
Inequality
Simplified
Solution Set
x+4 > 6
x > 2
x>2
x+6 > 4
x > -2
x>-2
4+6 > x
10 > x
x<10
Now we can graph the inequalities and find the set of common solutions.
As we can see, the lines overlap on the segment from 2 to 10. Therefore, the possible values of x are the following.
2
