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This module introduces rational expressions in equations.

Rational equations

A rational equation means that you are setting two rational expressions equal to each other. The goal is to solve for x; that is, find the x value(s) that make the equation true.

Suppose I told you that:

x 8 = 3 8 size 12{ { {x} over {8} } = { {3} over {8} } } {}

If you think about it, the x in this equation has to be a 3. That is to say, if x=3 then this equation is true; for any other x value, this equation is false.

This leads us to a very general rule.

A very general rule about rational equations

If you have a rational equation where the denominators are the same, then the numerators must be the same.

This in turn suggests a strategy: find a common denominator, and then set the numerators equal.

Example: Rational Equation
3 x 2 + 12 x + 36 = 4x x 3 + 4x 2 12 x size 12{ { {3} over {x rSup { size 8{2} } +"12"x+"36"} } = { {4x} over {x rSup { size 8{3} } +4x rSup { size 8{2} } - "12"x} } } {} Same problem we worked before, but now we are equating these two fractions, instead of subtracting them.
3 ( x ) ( x 2 ) ( x + 6 ) 2 ( x ) ( x 2 ) = 4x ( x + 6 ) x ( x + 6 ) 2 ( x 2 ) size 12{ { {3 \( x \) \( x - 2 \) } over { \( x+6 \) rSup { size 8{2} } \( x \) \( x - 2 \) } } = { {4x \( x+6 \) } over {x \( x+6 \) rSup { size 8{2} } \( x - 2 \) } } } {} Rewrite both fractions with the common denominator.
3 x ( x - 2 ) = 4 x ( x + 6 ) Based on the rule above—since the denominators are equal, we can now assume the numerators are equal.
3 x 2 6 x = 4 x 2 + 24 x Multiply it out
x 2 + 30 x = 0 What we’re dealing with, in this case, is a quadratic equation. As always, move everything to one side...
x ( x + 30 ) = 0 ...and then factor. A common mistake in this kind of problem is to divide both sides by x ; this loses one of the two solutions.
x= 0 or x= -30 Two solutions to the quadratic equation. However, in this case, x = 0 is not valid, since it was not in the domain of the original right-hand fraction. (Why?) So this problem actually has only one solution, x = 30 .

As always, it is vital to remember what we have found here. We started with the equation 3 x 2 + 12 x + 36 = 4x x 3 + 4x 2 12 x size 12{ { {3} over {x rSup { size 8{2} } +"12"x+"36"} } = { {4x} over {x rSup { size 8{3} } +4x rSup { size 8{2} } - "12"x} } } {} . We have concluded now that if you plug x = 30 into that equation, you will get a true equation (you can verify this on your calculator). For any other value, this equation will evaluate false.

To put it another way: if you graphed the functions 3 x 2 + 12 x + 36 size 12{ { {3} over {x rSup { size 8{2} } +"12"x+"36"} } } {} and 4x x 3 + 4x 2 12 x size 12{ { {4x} over {x rSup { size 8{3} } +4x rSup { size 8{2} } - "12"x} } } {} , the two graphs would intersect at one point only: the point when x = 30 .

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
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?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
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?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Math 1508 (lecture) readings in precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11354/1.1
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