In the triangle shown, for \(\angle A\) to be the largest angle of the triangle, it must be that m < x < n. What is the least possible value of \(n-m\),expressed as a common fraction?
We have that either
x + 4 + 3x > x + 9 or 3x + x + 9 > x + 4 or x + 4 + x + 9 > 3x
4x + 4 > x + 9 4x + 9 > x + 4 2x + 13 > 3x
3x > 5 3x > -5 13 > x
x > 5/3 x > -5/3 x < 13
Taking the most restrictive interval
5/3 < x < 13
So
n - m = 13 - 5/3 = 39/3 - 5/3 = 34 / 3
+1206 
