Solving linear equations and inequalities: Linear inequalities in one unknown
Reduction to a linear inequality
In some cases, you can reduce complicated inequalities to linear inequalities.
We note first that division by zero is not allowed and that for this reason \(6x+3\) may not be equal to zero and that therefore \(x=-{{1}\over{2}}\) is not a solution.
We now distinguish two cases, namely \(6x+3>0\) and \(6x+3<0\).
In both cases we multiply the inequality on both sides by \(6x+3\) because we then get a linear inequality, for which we know there is a solution method.
Suppose \(6x+3>0\), i.e. \(x> -{{1}\over{2}}\). Then we get \(7<9(6x+3)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(-54x<20\).
Then, dvision by the coefficient of \(x\)gives \(x > -{{10}\over{27}}\).
So we have the following system of inequalities: \(x> -{{1}\over{2}}\,\wedge\; x > -{{10}\over{27}}\)
and this simplifies to \(x\gt-{{10}\over{27}}\).
Suppose \(6x+3<0\), i.e. \(x< -{{1}\over{2}}\). Then we get \(7>9(6x+3)\).
When we move everything with \(x\) to the left and all constant terms to the right, we get \(-54x>20\).
Then, division by the coefficient of \(x\) gives \(x < -{{10}\over{27}}\).
So we have the following system of inequalities: \(x< -{{1}\over{2}}\,\wedge\; x < -{{10}\over{27}}\)
and this simplifies to \(x\lt -{{1}\over{2}}\).
The solution of the original inequality is \(x\lt -{{1}\over{2}}\;\vee\;x\gt-{{10}\over{27}}\).