We note first that division by zero is not allowed and that for this reason \(9x-9\) may not be equal to zero and that therefore \(x=1\) is not a solution.

We now distinguish two cases, namely \(9x-9>0\) and \(9x-9<0\).
In both cases we multiply the inequality on both sides by \(9x-9\) because we then get a linear inequality, for which we know there is a solution method.

Suppose \(9x-9>0\), i.e. \(x> 1\). Then we get \(9<-(9x-9)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(9x<0\).
Then, dvision by the coefficient of \(x\)gives \(x < 0\).
So we have the following system of inequalities: \(x> 1\,\wedge\; x < 0\)
and this simplifies to \(\text{an empty solution set}\).

Suppose \(9x-9<0\), i.e. \(x< 1\). Then we get \(9>-(9x-9)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(9x>0\).
Then, division by the coefficient of \(x\) gives \(x > 0\).
So we have the following system of inequalities: \(x< 1\,\wedge\; x > 0\)
and this simplifies to \(0\lt x\lt 1\).

The solution of the original inequality is \(0\lt x\lt 1\).