We are given three expressions for measures of the sides of a triangle.
x+1, 5, 7
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+1+ 5> 7
(II):& x+1+ 7> 5
(III):& 5+ 7> x+1
Let's solve each inequality for x.
Inequality
Simplified
Solution Set
x+1+5 > 7
x > 1
x>1
x+1+7 > 5
x > -3
x>-3
5+7 > x+1
11 > x
x<11
Now we can graph the inequalities and find the set of common solutions.
As we can see, the lines overlap on the segment from 1 to 11. Therefore, the possible values of x are the following.
1
