The way you have it, " a " could be, say, -100000 and " b " could be, say,
100000.  

I think you want to find the largest possible value for a and the 
smallest possible value for b such that 2 < x < 4 and a < [1/(x - 6)] < b

As x approaches 2 from the right, (going left)  approaches ,

So one of a and b is -1/4

As x approaches 4 from the left, (going right)  approaches , 

So the other one of a and b is -1/2

Since a is less than b, a = -1/2 and b = -1/4

So we have 

with  and 

Edwin