Polynomial and Rational Inequalities
Introduction
The methods for solving polynomial and rational inequalities are very similar. The main difference is that we have to be aware of vertical asymptotes when considering rational inequalities.
Polynomial Inequalities
Step 1: Put all nonzero terms on the same side, preferably which ever side leads to a positive leading coefficient.
Step 2: Factor the polynomial as far as possible
Step 3: Find the zeros of the polynomial
Step 4: Use the zeros to divide the real number line into regions.
Step 5: Determine over which regions the polynomial is positive and which regions the polynomial is negative
Step 6: Write the desired intervals based on the inequality.
Rational Inequalities
The main difference here is during step 3 you want to find the zeros of both numerator and denominator. Then when you divide the real number line you want to account for the vertical asymptotes and zeros of the function.