SOLUTION: How would I solve this rational inequality: (x/x-6)<2 Thank you

 < 2.   (1)


1. Assume that x - 6 > 0, i.e. x > 6.
   
   Multiply both sides of (1) by (x-6), which is positive in this case. You will get an inequality

   x < 2*(x-6)  --->  x < 2x - 12  --->  12 > x.

   Thus the solution in this case is the set of real {x | 6 < x < 12}, i.e the interval (6,12).


2. Assume that x - 6 < 0, i.e. x < 6.
   
   Multiply both sides of (1) by (x-6), which is negative in this case. You will get an inequality

   x > 2*(x-6)  --->  x > 2x - 12  --->  12 > x.   <----  Notice that I changed the inequality sign when multiplied by negative number!

   Thus the solution in this case is the set of real {x | x < 6}, i.e the semi-infinite interval (,6).

Answer. The solution is the union of two intervals: (,6) U {6,12}.

Figure. Plot y =