You need to remember that when you multiply or divide by negative numbers, the inequality sign changes its direction. So you will need to consider several different cases.
we have the inequality x/(6x-9) <= 1/x , if we get rid of the denominators we get
x^2 <= 6x-9 , x!= 3/2 and 0
x^2-6x+9<=0
(x-3)^2 <=0
So the solution would be only at x=3. However the real solution includes 0<x<1.5 , but this does not work here? Now if we just subtract 1/x from x/(6x-9), it will give us the correct answer.
What did I do wrong, am I missing something in my approach?