Here are the steps to solve polynomial inequalities:

1. Move everything to the left hand side to  make your inequality looks like (here the inequality symbol can also be or )
2. Find the zeros of (by factoring if it is possible)
3. Plot the zeros on the number line and these zeros will divide the number line into several intervals
4. Pick a sample point on each interval and plugin it to
5. If after plugin you get is true then the interval on the number line where your belong is part of the solution, otherwise the interval is not part of the solution

The videos below will guide you step by step to solve te polynomial inequalities using a concrete example

The idea to solve rational inequalities is first to rewrite the inequality so that the left hand side is a single rational function and the right hand side is zero. So the firs step is to rewrite the inequality so it looks like (of course here the inequality symbol can be or ). The next step is to find for which the numerator is zero and for which 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 . If you get a right statement after you plugin your 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 , then you consider two cases. the case when and the case when . If then multiplying the inequality by we have (remember we need to reverse the inequality sign since is negative) and then solve the inequality . If , multiplying both sides of the inequality by we have and then we can solve this inequality.

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