We note first that division by zero is not allowed and that for this reason \(7x+9\) may not be equal to zero and that therefore \(x=-{{9}\over{7}}\) is not a solution.

We now distinguish two cases, namely \(7x+9>0\) and \(7x+9<0\).
In both cases we multiply the inequality on both sides by \(7x+9\) because we then get a linear inequality, for which we know there is a solution method.

Suppose \(7x+9>0\), i.e. \(x> -{{9}\over{7}}\). Then we get \(4<-(7x+9)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(7x<-13\).
Then, dvision by the coefficient of \(x\)gives \(x < -{{13}\over{7}}\).
So we have the following system of inequalities: \(x> -{{9}\over{7}}\,\wedge\; x < -{{13}\over{7}}\)
and this simplifies to \(\text{an empty solution set}\).

Suppose \(7x+9<0\), i.e. \(x< -{{9}\over{7}}\). Then we get \(4>-(7x+9)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(7x>-13\).
Then, division by the coefficient of \(x\) gives \(x > -{{13}\over{7}}\).
So we have the following system of inequalities: \(x< -{{9}\over{7}}\,\wedge\; x > -{{13}\over{7}}\)
and this simplifies to \(-{{13}\over{7}}\lt x\lt -{{9}\over{7}}\).

The solution of the original inequality is \(-{{13}\over{7}}\lt x\lt -{{9}\over{7}}\).