Solving rational inequalities?

Question

(x-2)/x < (x-4)/(x-6)

 

How to get an answer of 0<x<3 or x>6

Answer 1

In case you are wondering where the 3 comes from:  Subtract quantity (x-4/x-6) from both sides to get 0 on the right side. You must get 0 there. Get LCD of x(x-6). Simplify numerator to -4x+12 and factor it to -4(x-3). Ah!  There's the 3. Now both numerator and denominator are in factored form. Find the zeros in both num and den. These are the critical points. Choose a value in each segment and test it in the original inequality to find which intervals "work".  You will see that between 0 and 3 "works". Also anything greater than 6 "works".  Hope this helps.  And nothing else does.

Answer 2

(x-2)/x<(x-4)/(x-6)--->(x-2)/x-(x-4)/(x-6)<0--->-4(x-3)/(x(x-6))<0---->critical points are 0, 3, 6,

Test values around critical points to get the final answer.