We note first that division by zero is not allowed and that for this reason \(8x+7\) may not be equal to zero and that therefore \(x=-{{7}\over{8}}\) is not a solution.

We now distinguish two cases, namely \(8x+7>0\) and \(8x+7<0\).
In both cases we multiply the inequality on both sides by \(8x+7\) because we then get a linear inequality, for which we know there is a solution method.

Suppose \(8x+7>0\), i.e. \(x> -{{7}\over{8}}\). Then we get \(7<-4(8x+7)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(32x<-35\).
Then, dvision by the coefficient of \(x\)gives \(x < -{{35}\over{32}}\).
So we have the following system of inequalities: \(x> -{{7}\over{8}}\,\wedge\; x < -{{35}\over{32}}\)
and this simplifies to \(\text{an empty solution set}\).

Suppose \(8x+7<0\), i.e. \(x< -{{7}\over{8}}\). Then we get \(7>-4(8x+7)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(32x>-35\).
Then, division by the coefficient of \(x\) gives \(x > -{{35}\over{32}}\).
So we have the following system of inequalities: \(x< -{{7}\over{8}}\,\wedge\; x > -{{35}\over{32}}\)
and this simplifies to \(-{{35}\over{32}}\lt x\lt -{{7}\over{8}}\).

The solution of the original inequality is \(-{{35}\over{32}}\lt x\lt -{{7}\over{8}}\).