We note first that division by zero is not allowed and that for this reason \(3x+4\) may not be equal to zero and that therefore \(x=-{{4}\over{3}}\) is not a solution.

We now distinguish two cases, namely \(3x+4>0\) and \(3x+4<0\).
In both cases we multiply the inequality on both sides by \(3x+4\) because we then get a linear inequality, for which we know there is a solution method.

Suppose \(3x+4>0\), i.e. \(x> -{{4}\over{3}}\). Then we get \(1<9(3x+4)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(-27x<35\).
Then, dvision by the coefficient of \(x\)gives \(x > -{{35}\over{27}}\).
So we have the following system of inequalities: \(x> -{{4}\over{3}}\,\wedge\; x > -{{35}\over{27}}\)
and this simplifies to \(x\gt-{{35}\over{27}}\).

Suppose \(3x+4<0\), i.e. \(x< -{{4}\over{3}}\). Then we get \(1>9(3x+4)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(-27x>35\).
Then, division by the coefficient of \(x\) gives \(x < -{{35}\over{27}}\).
So we have the following system of inequalities: \(x< -{{4}\over{3}}\,\wedge\; x < -{{35}\over{27}}\)
and this simplifies to \(x\lt -{{4}\over{3}}\).

The solution of the original inequality is \(x\lt -{{4}\over{3}}\;\vee\;x\gt-{{35}\over{27}}\).