Solving Polynomial and Rational Inequalities
Put the inequality in standard form.
To do this collect all terms on one side.
If you have fractions obtain a single fraction by adding / subtracting all fractions.
Do not clear any denominators that contain variables.
Polynomial case: Left side is a polynomial
(1) Find all zeros of by solving the equation
These are called the critical numbers.
(2) Place the critical numbers found on a real number line. These numbers will split the real number line into a number of intervals.
(3) Make a “table of signs”
To do this pick test numbers (other than the critical numbers) in each interval found above and find the sign of in that interval by evaluating at the corresponding test number. Record that sign on the table.
Note: The sign of is the same throughout the interval so it is enough to check the sign at a single test number.
(4) Obtain the solution, check endpoints.
Collect the intervals for which the sign found as above is as desired.
If the inequality includes the equal sign (non-strict inequality) then you should include the endpoints of the intervals.
Find out more about solving polynomials and rational inequalities below.
Solving Polynomial and Rational Inequalities
Practice: Solve second degree inequalities in the following exercises.