We note first that division by zero is not allowed and that for this reason \(x-8\) may not be equal to zero and that therefore \(x=8\) is not a solution.

We now distinguish two cases, namely \(x-8>0\) and \(x-8<0\).
In both cases we multiply the inequality on both sides by \(x-8\) because we then get a linear inequality, for which we know there is a solution method.

Suppose \(x-8>0\), i.e. \(x> 8\). Then we get \(4<-9(x-8)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(9x<68\).
Then, dvision by the coefficient of \(x\)gives \(x < {{68}\over{9}}\).
So we have the following system of inequalities: \(x> 8\,\wedge\; x < {{68}\over{9}}\)
and this simplifies to \(\text{an empty solution set}\).

Suppose \(x-8<0\), i.e. \(x< 8\). Then we get \(4>-9(x-8)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(9x>68\).
Then, division by the coefficient of \(x\) gives \(x > {{68}\over{9}}\).
So we have the following system of inequalities: \(x< 8\,\wedge\; x > {{68}\over{9}}\)
and this simplifies to \({{68}\over{9}}\lt x\lt 8\).

The solution of the original inequality is \({{68}\over{9}}\lt x\lt 8\).