Question 898933
.


            The solution by  @CubeyThePenguin is  INCORRECT.


            I came to bring you a correct solution.



<pre>
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 < {{{1/(9x-27)}}};  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/(9x-27)}}} < 1/9;  it is equivalent to  {{{1/(x-3)}}} < 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.


<U>ANSWER</U>.  The solution set is  x > 4.


                         <U>Visual CHECK</U>



    {{{graph( 400, 400, -10, 10, -1.5, 1.5, 
              1/(9x-27), 1/9
)}}}   


      Figure.  Plot y = {{{1/(9x-27)}}} (red)  and y = {{{1/9}}} (green)
</pre>

Solved &nbsp;(correctly).