Elementary Algebra
Introduction to Algebra Linear Equations and Inequalities Functions and Graphs I Lines and thier Graphs Linear Systems Exponents & Polynomials
Intermediate Algebra
Factoring Rational Expressions Rational Equations and Applications Radical Expressions Nonlinear Equations and Applications Functions and Graphs II Exponential and Logarithmic Functions
Precalculus I / College Algebra
Equations and Inequalitites Functions and Graphs Polynomial and Rational Functions Exponential and Logarithmic Functions Systems and Matrices Geometry Basics Conic Sections Sequences and Series
Precalculus II / Trigonometry
The Six Trigonometric Functions Right Triangle Trigonometry Circular Functions Graphs of Trigonometric Functions Trigonometric Identities Trigonometric Equations Oblique Triangles and the Laws Vectors Complex, Parametric, and Polar Forms
Calculus I
Limits and Continuity Derivatives Analysis of Curves Antiderivatives
Calculus II
Transcendental Functions
Geometry Physics Integration Techniques Calculus of Infinity Parametric, Polar, and Conic Curves
Calculus III

Course: Intermediate Algebra
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

Terms you should be able to 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.