# 8.6 Rational equations

 Page 1 / 2
<para>This module is from<link document="col10614">Elementary Algebra</link>by Denny Burzynski and Wade Ellis, Jr.</para><para>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.</para><para>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.</para><para>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.</para>

## Overview

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

## 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):

$\frac{3x}{4}=\frac{15}{2}$

$\frac{x+1}{x-2}=\frac{x-7}{x-3}$

$\frac{5a}{2}=10$

$\frac{3}{x}+\frac{x-3}{x+1}=\frac{6}{5x}$

$\frac{x-6}{x+1}$ is a rational expression , not a rational equation.

## 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.

$12\left(\frac{1}{6}+\frac{x}{4}\right)=12\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{17}{12}$

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

$12\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{6}+12\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{x}{4}=12\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{17}{12}$

Now divide to eliminate all denominators.

$\begin{array}{ccc}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}1+3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}x& =& 17\\ \hfill 2+3x& =& 17\end{array}$

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

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.

## Clearing an equation of fractions

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

what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform. By By  By By  By By  By By Mldelatte By