How To Solve Rational Inequalities

The idea to solve rational inequalities is first to rewrite the inequality so that the left hand side is a single rational function $\frac{p(x)}{q(x)}$ and the right hand side is zero. So the firs step is to rewrite the inequality so it looks like $\frac{p(x)}{q(x)}<0$ (of course here the inequality symbol can be $>,\leq$ or $\geq$ ). The next step is to find for which $x$ the numerator is zero and for which $x$ the denominator is zero. Plot all these zeros on a number line. These zeros will divide the number line into several subintervals. Take a sample point on each sub interval and plugin to $\frac{p(x)}{q(x)}>0$ . If you get a right statement after you plugin your $x$ to the inequality, that means the subinterval is a solution to the inequality.

Another method is that after you rewrite your inequality so that it has the form $\frac{p(x)}{q(x)}<0$ , then you consider two cases. the case when $q(x)<0$ and the case when $q(x)>0$ . If $q(x)<0$ then multiplying the inequality by $q(x)$ we have $p(x)>0$ (remember we need to reverse the inequality sign since $q(x)$ is negative) and then solve the inequality $p(x)>0$ . If $q(x)>0$ , multiplying both sides of the inequality by $q(x)$ we have $p(x)<0$ and then we can solve this inequality.

The video belows will guide you to use these to methods to solve rational inequalities.