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)
