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?