Chapter 8: Rational and Radical Functions

Section 5: Solving Rational Equations and Inequalities

Objectives

Solve rational equations and inequalities.

Vocabulary

rational equation extraneous solution rational inequality

Who Uses This?

Kayakers can use rational equations to determine how fast a river is moving.

(See Example 3.)

8.5: Solving Rational Equations and Inequalities A rational equation is an

equation that contains one or more rational expressions.

The time t in hours that it takes to travel d miles can be determined by using the equation t = d/r , where r is the average rate of speed.

This equation is a rational equation.

8.5: Solving Rational Equations and Inequalities To solve a rational equation, start by

multiplying each term of the equation by the least common denominator (LCD) of all of the expressions in the equation.

This step eliminates the denominators of the rational expressions and results in an equation you can solve by using algebra.

Example 1: Solving Rational Equations

8.5: Solving Rational Equations and Inequalities An extraneous solution is a

solution of an equation derived from an original equation that is not a solution of the original equation.

When you solve a rational equation, it is possible to get extraneous solutions.

These values should be eliminated from the solution set.

Always check your solutions by substituting them into the original equation.

Example 2: Extraneous Solutions

Example 3: Problem Solving Application

Example 4: Work Application

8.5: Solving Rational Equations and Inequalities A rational inequality is an

inequality that contains one or more rational expressions.

One way to solve rational inequalities is by using graphs and tables.

Example 5: Using Graphs and Tables to Solve Rational Equations and Inequalities

8.5: Solving Rational Equations and Inequalities You can also solve rational

inequalities algebraically. You start by multiplying each term

by the least common denominator (LCD) of all the expressions in the inequality.

However, you must consider two cases: the LCD is positive or the LCD is negative.

Example 6: Solving Rational Inequalities Algebraically

OTL

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