 A polynomial inequality is an inequality that can take on 1 of 4 forms:  f(x) < 0  f(x) > 0  f(x) ≤ 0  f(x) ≥ 0  Here the function f(x) is a polynomial function.

 There are 5 steps to solving polynomial inequalities.  Let’s look at an example and see these steps in action.

Solve the following polynomial inequality:

Where the equation crosses the x-axis. We call the points where the function crosses the x-axis the boundary points.

Intervals (-∞,-2) (-2, 3) (3,∞) The function does not touch the x-axis when within these intervals.

x

IntervalsTest ValueEvaluate Test ValueConclusion (-∞,-2)-5f(x) > 0 for all x in (-∞,-2) (-2, 3)0f(x) < 0 for all x in (-2, 3) (3,∞)4f(x) > 0 for all x in (3,∞)

The above polynomial inequality is only true when x is less than -2 or greater than 3. Solution set

 If the function has the sign, you do not include the boundary points in the solution set.  If the function has the ≤ or ≥ sign, you do include the boundary points in the solution set.