8.6 Rational equations

Overview

• Rational Equations
• The Logic Behind The Process
• The Process
• Extraneous Solutions

Rational equations

Rational equations

When one rational expression is set equal to another rational expression, a rational equation results.

Some examples of rational equations are the following (except for number 5):

The logic behind the process

It seems most reasonable that an equation without any fractions would be easier to solve than an equation with fractions. Our goal, then, is to convert any rational equation to an equation that contains no fractions. This is easily done.

To develop this method, let’s consider the rational equation

$$\frac{1}{6}+\frac{x}{4}=\frac{17}{12}$$

The LCD is 12. We know that we can multiply both sides of an equation by the same nonzero quantity, so we’ll multiply both sides by the LCD, 12.

$$12\left(\frac{1}{6}+\frac{x}{4}\right)=12·\frac{17}{12}$$

Now distribute 12 to each term on the left side using the distributive property.

$$12·\frac{1}{6}+12·\frac{x}{4}=12·\frac{17}{12}$$

Now divide to eliminate all denominators.

$$\begin{array}{ccc}2·1+3·x& =& 17\\ \hfill 2+3x& =& 17\end{array}$$

Now there are no more fractions, and we can solve this equation using our previous techniques to obtain 5 as the solution.

The process

We have cleared the equation of fractions by multiplying both sides by the LCD. This development generates the following rule.

Clearing an equation of fractions

When multiplying both sides of the equation by the LCD, we use the distributive property to distribute the LCD to each term. This means we can simplify the above rule.

Clearing an equation of fractions