Topic: Rational Equations and Applications

Subtopic: Rational Equations

**Overview**

Solving rational equations is the focus of this lesson. It is important to recognize the difference between a rational equation and a rational expression. With equations we can actually solve for x and since there is an equals sign we are able to multiply both sides by something to eliminate the denominators. This wasn't true with expressions. When working with rational expressions all we could do, for example, is add them together by building up the denominators to be the same -- you could not multiply through by the LCD and eliminate it.

We will solve two types of rational equations. __Proportions__ (single fraction = single fraction) are solved by the __cross products method__. More complicated rational expressions, ones containing more than two terms, are solved by the __LCD method__ where the entire expression is multiplied through by the LCD to clear all the denominators.

**Objectives**

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

- 9.1.1 Know the difference between a rational expression and a rational equation
- 9.1.2 Recognize a proportion
- 9.1.3 Solve rational equations that are proportions by the cross products method
- 9.1.4 Find the LCD of a rational equation
- 9.1.5 Solve rational equations that are not proportions by the LCD method
- 9.1.6 Find domain restrictions of rational equations
- 9.1.7 Recognize when an apparent solution to a rational equation is actually a domain restriction to the original equation

**Terminology**

Define: rational equation, proportion, cross products method (a.k.a. means and extremes)

**Text Notes**

When solving a rational equation, concentrate on the algebraic approach. Solving graphically is covered more in a pre-calculus class. However, you may use the graphing method (electronically graphing each side of the equation as a separate function and finding the intersection points) to check your answers if you would like.

When finding the domain restriction, concentrate on the algebraic approach (finding the x-values that make the denominators in the original equation zero) not the graphical approach (which represents domain restrictions of rational expressions as vertical asymptote lines). Always make sure that you do not include any domain restrictions in your final list of solutions.