You start from this inequality


    0 < (9x-27)^(-1) < 1/9


It is equivalent to the system of two inequalities


    0 < (9x-27)^(-1)          (1)
and
        (9x-27)^(-1) < 1/9    (2)


connected with the service word "and".


First inequality (1)  is  0 < ;  it is equivalent to  9x - 27 > 0;  x - 3 > 0  and finally  x > 3.


        So for what is follows, we can assume x > 3.     (3)


Second inequality (2)  is   < 1/9;  it is equivalent to   < 1  and finally 

(taking into account that x > 3) it implies  1 < x-3; hence x > 4.   (4)


Thus we get the final answer  x > 4.


ANSWER.  The solution set is  x > 4.


                         Visual CHECK



       


      Figure.  Plot y =  (red)  and y =  (green)