We note first that division by zero is not allowed and that for this reason \(6x+1\) may not be equal to zero and that therefore \(x=-{{1}\over{6}}\) is not a solution.

We now distinguish two cases, namely \(6x+1>0\) and \(6x+1<0\).
In both cases we multiply the inequality on both sides by \(6x+1\) because we then get a linear inequality, for which we know there is a solution method.

Suppose \(6x+1>0\), i.e. \(x> -{{1}\over{6}}\). Then we get \(4<6x+1\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(-6x<-3\).
Then, dvision by the coefficient of \(x\)gives \(x > {{1}\over{2}}\).
So we have the following system of inequalities: \(x> -{{1}\over{6}}\,\wedge\; x > {{1}\over{2}}\)
and this simplifies to \(x\gt{{1}\over{2}}\).

Suppose \(6x+1<0\), i.e. \(x< -{{1}\over{6}}\). Then we get \(4>6x+1\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(-6x>-3\).
Then, division by the coefficient of \(x\) gives \(x < {{1}\over{2}}\).
So we have the following system of inequalities: \(x< -{{1}\over{6}}\,\wedge\; x < {{1}\over{2}}\)
and this simplifies to \(x\lt -{{1}\over{6}}\).

The solution of the original inequality is \(x\lt -{{1}\over{6}}\;\vee\;x\gt{{1}\over{2}}\).