In a triangle with side lengths 5, 6 and $x$, what is the sum of all possible integral values of $x$?
+128407 5 , 6 and x
By the triangle inequality, we have two situations
We have that
5 + 6 > x
11 > x
So x < 11
And
5 + x > 6
x > 1
So...we have the following dual inequality
1 < x < 11
So....the possible integer values for x are 2,3,4,5,6,7,8,9,10
And the sum of these is just (10) (11) / 2 - 1 = 55 - 1 = 54
