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

Inside Collection (Textbook): Elementary Algebra

Summary:

This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr.

A detailed study of arithmetic operations with rational expressions is presented in this chapter, beginning with the definition of a rational expression and then proceeding immediately to a discussion of the domain. The process of reducing a rational expression and illustrations of multiplying, dividing, adding, and subtracting rational expressions are also included. Since the operations of addition and subtraction can cause the most difficulty, they are given particular attention. We have tried to make the written explanation of the examples clearer by using a "freeze frame" approach, which walks the student through the operation step by step.

The five-step method of solving applied problems is included in this chapter to show the problem-solving approach to number problems, work problems, and geometry problems. The chapter also illustrates simplification of complex rational expressions, using the combine-divide method and the LCD-multiply-divide method.

Objectives of this module: be able to identify rational equations, understand and be able to use the method of solving rational expressions, be able to recognize extraneous solutions.

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

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):

*expression*, not a rational equation.

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

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.

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

Now divide to eliminate all denominators.

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

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

To clear an equation of fractions, multiply both sides of the equation by the LCD.

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.

To clear an equation of fractions, multiply *every* term on both sides of the equation by the LCD.

The complete method for solving a rational equation is

1. Determine all the values that must be excluded from consideration by finding the values that will produce zero in the denominator (and thus, division by zero). These excluded values are not in the domain of the equation and are called nondomain values.

2. Clear the equation of fractions by multiplying every term by the LCD.

3. Solve this nonfractional equation for the variable. Check to see if any of these potential solutions are excluded values.

4. Check the solution by substitution.

Potential solutions that have been excluded because they make an expression undefined (or produce a false statement for an equation) are called extraneous solutions. Extraneous solutions are discarded. If there are no other potential solutions, the equation has no solution.

Solve the following rational equations.

Solve the following rational equations.

Solve the following rational equations.

The zero-factor property can be used to solve certain types of rational equations. We studied the zero-factor property in Section 7.1, and you may remember that it states that if

Solve the equation

Solve the equation

This equation has no solution.

For the following problems, solve the rational equations.

No solution; 6 is an excluded value.

no solution

No solution;

For the following problems, solve each literal equation for the designated letter.

*((Reference))* Write

*((Reference))* Factor

*((Reference))* Supply the missing word. An slope of a line is a measure of the

of the line.

steepness

*((Reference))* Find the product.

*((Reference))* Find the sum.

- « Previous module in collection Adding and Subtracting Rational Expressions
- Collection home: Elementary Algebra
- Next module in collection » Applications

Comments:"Reviewer's Comments: 'I recommend this book for courses in elementary algebra. The chapters are fairly clear and comprehensible, making them quite readable. The authors do a particularly nice job […]"