Algebra and Trigonometry, 5th Edition

4.6 Polynomial Inequalities and Rational Inequalities

• Solve polynomial inequalities.

• Solve rational inequalities.

We will use a combination of algebraic methods and graphical methods to solve polynomial inequalities and rational inequalities.

Polynomial Inequalities

Just as a quadratic equation can be written in the form $a{x}^2+bx+c=0$, a quadratic inequality can be written in the form $a{x}^2+bx+c0$, where is $<,>,\le$, or $\ne$. Here are some examples of quadratic inequalities:

$$x^2-4x-5<0\text{and}-\frac{1}{2}{x}^2+4x-7\ge 0.$$

When the inequality symbol in a polynomial inequality is replaced with an equals sign, a related equation is formed. Polynomial inequalities can be solved once the related equation has been solved.

Example 1

Solve: $x^2-4x-5>0$.

Solution

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