Quadratic and Other Polynomial Inequalities
Inequalities like the following are called quadratic inequalities:
x2 + 3x – 10 < 0, 5x2 – 3x + 2 ≥ 0.
In each case, we have a polynomial of degree 2 on the left. We will solve such inequalities in two ways. The first method provides understanding and the second yields the more efficient method.
The first method for solving a quadratic inequality, such as ax2 + bx + c > 0, is by considering the graph of a related function, f(x) = ax2 + bx + c.
To solve a polynomial inequality:
a. Get 0 on one side, set the expression on the other side equal to 0,
and solve to find the x-intercepts.
b. Use the numbers found in step (a) to divide the number line into
intervals.
c. Substitute a number from each interval into the related function.
If the function value is positive, then the expression will be
positive for all numbers in the interval. If the function value is
negative, then the expression will be negative for all numbers in
the interval.
d. Select the intervals for which the inequality is satisfied and write
set-builder or interval notation for the solution set.