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

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

Suppose \(x+7>0\), i.e. \(x> -7\). Then we get \(5<8(x+7)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(-8x<51\).
Then, dvision by the coefficient of \(x\)gives \(x > -{{51}\over{8}}\).
So we have the following system of inequalities: \(x> -7\,\wedge\; x > -{{51}\over{8}}\)
and this simplifies to \(x\gt-{{51}\over{8}}\).

Suppose \(x+7<0\), i.e. \(x< -7\). Then we get \(5>8(x+7)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(-8x>51\).
Then, division by the coefficient of \(x\) gives \(x < -{{51}\over{8}}\).
So we have the following system of inequalities: \(x< -7\,\wedge\; x < -{{51}\over{8}}\)
and this simplifies to \(x\lt -7\).

The solution of the original inequality is \(x\lt -7\;\vee\;x\gt-{{51}\over{8}}\).