Quadratic and Power Inequalities
A Quadratic Inequality is an inequality that can be written in the form: ax2
+ bx + c > 0 where a, b, and c are real numbers with a
≠ 0. (or with > replaced by <,
≤, or ≥)
We will solve quadratic inequlities both algebraically and graphically. Algebraically, we use a method which
involves factoring and a sign diagram. The steps are below.
Algebraic Method for Solving Polynomial Inequalities
1. Write the inequality so that a polynomial expression f(x) is
on the left side and zero is on the right side in one of these forms:
|f(x) > 0
||f(x) < 0
||f(x) ≤ 0
2. Determine the numbers at which the expression f(x) on the left
side equals zero. (That is, solve f(x) = 0.) These numbers are the Critical Numbers.
3. Mark on a number line the intervals determined by all of the
4. Use a Sign Diagram to determine which intervals are in the
solution set in the following manner:
(a) Select a in each of the intervals and evaluate f(x) test number
at the test number.
(b) If the value of f(x) is positive, then f(x) > 0 for all numbers x in the interval.
(c) If the value of f(x) is negative, then f(x) < 0 for all numbers x in the interval.
5. Give the answer either in inequality form or in interval notation.
Now let's try the same problem graphically. Again the first step is to rewrite the inequality so that zero is on the
right. Then we graph Y1 = f(x). We then locate the zeros of f(x), that is, the x-intercepts.
The solution to the inequality involves an interval or intervals on the x-axis where the function values are
greater than or equal to 0. Where does that happen on this graph?