To solve a polynomial inequality, first put a zero on one side of the inequality. Then, factor the inequality if it is factorable, although this attribute applies to only some inequalities. The third step is to calculate the x-intercepts of the inequality, which are points where the inequality is zero. The next step is to graph the zero points of the inequality.

The most important step in finding the solution to polynomial inequalities is to find the x-intercepts, as it is at these points where the sign of an inequality can change. Since the graphs of polynomial inequalities are smooth curves, they typically move along the number line and must pass through zero to change signs. This means that if the inequality is satisfied at one point in a specific region, all the points in that region satisfy the inequality.

Another step is to graph the points where the polynomial is zero on a number line. Once this is done, choose any value of X from the different regions and place it in the factored inequality to determine the sign of the inequality at that point. Lastly, write down the solution to the inequality from the required regions in interval notation.