In order for ∠A to be the largest angle, It must be that the side formed by ∠A is the largest side. So...
| 3x < x + 9 | |
| 2x < 9 |
|
| x < \(\frac92\) |
But by the triangle inequality theorem,
| x + 9 < ( 3x ) + ( x + 4 ) | |
| x + 9 < 3x + x + 4 |
|
| 9 < 3x + 4 | |
| 5 < 3x |
|
| \(\frac53\) < x |
So in order for ∠A to be the largest angle, it must be that \(\frac53
The least possible value of n - m is \(\frac92-\frac53\ =\ \frac{27}{6}-\frac{10}{6}\ =\ \boxed{\frac{17}{6}}\)
( Here's an older answer for this question: https://web2.0calc.com/questions/geometry-question. . .#r5 )