Topic: Equations and Inequalities

Subtopic: Polynomial and Rational Inequalities

**Overview**

In Calculus I you are required to solve inequalities on f(x), f'(x) (the first derivative of f), and f"(x) (the second derivative of f) all in one problem. Since f(x) and its derivatives could be linear, polynomial, rational, or otherwise, it is helpful to be proficient in solving inequalities involving a variety of functions.

The main goal is to determine when the function is positive and when it is negative. This can be accomplished graphically or algebraically. If algebraically, the __sign chart method__ of solving these inequalities is the most efficient and reliable, but most texts use a __test point method__ instead. Either works and we can compare and contrast the methods in class.

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 1.3.1 Solve polynomial and rational inequalities graphically
- 1.3.2 Solve polynomial and rational inequalities algebraically by the sign chart method or the test point method
- 1.3.3 Recognize inequalities that have "no solution" or "all Real solutions"
- 1.3.4 Write solutions to inequalities in interval notation (the most common way in Calculus), inequality notation, set-builder notation, and graph solutions on a number line

**Terminology**

Define: polynomial inequality, rational inequality, sign chart, test point

**Text Notes**

Practice solving a variety of polynomial and rational inequalities both algebraically and graphically. Concentrate on the pure process without having to expand to applications.

You may SKIP any "curve fitting" and "regression" examples/problems in this section and throughout the course.

**Supplemental Resources (optional)**

If you need supplemental tutorial videos with examples relevant to this section go to James Sousa's MathIsPower4U and search for topics:

"Solving Polynomial Inequality"

"Solving Rational Inequalities"