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f 2 = 2 2 2 2 2 2 3 2 8 = 4 + 2 1 4 + 6 8 = 5 2 > 0

We summarize steps for drawing sign scheme/ diagram as :

1: Decompose both numerator and denominator into linear factors. Do not cancel common linear factors. Find critical points by equating linear factors individually to zero.

2: Mark distinct critical points on a real number line. If n be the numbers of distinct critical points, then real number line is divided into (n+1) sub-intervals.

3: Test sign of function in a particular interval. Assign alternate signs in adjacent sub-intervals.

4: If a linear factor is repeated even times, then sign of function will not alternate about the critical point corresponding to linear factor in question.

Solution of rational inequalities using sign scheme or diagram

An important point about interpreting sign diagram is that sign of function relates to non-zero values of function. Note that zero does not have sign. The critical points corresponding to numerator function are zeroes of rational function. As such, the graph of function is continuous at these critical points and these critical points can be included in the sub-interval. On the other hand, the rational function is not defined for critical points corresponding to denominator function (as denominator turns zero). We, therefore, conclude that an interval can include critical points corresponding to numerator function, but not the critical points corresponding to denominator function. In case, there are common critical points between numerator and denominator, then those critical points can not be included in the sub-interval.

We can interpret sign diagram in two ways. Either we determine the solution of a given quadratic inequality or we determine intervals of all four types of inequalities for a given quadratic expression. We shall illustrate these two approaches by working with the example case.

Determining solution of a given quadratic inequality

Let us consider that we are required to solve rational inequality

f x = x 2 x 2 x 2 3 x 8 0

The sign diagram as drawn earlier for the given rational function is shown here :

Sign scheme/ diagram

Sign scheme/ diagram

We need to interpret signs of different intervals to find the solution of a given rational inequality.

Clearly, solution of given inequality is :

x ( - , 2 ] U 4, { 1 , 4 }

Note that we need to remove -1 and 4 from the solution set as function is not defined for this x – value. However, inequality involved “greater than or equal to” is not strict inequality. It allows equality to zero. As such, we include critical point “2” belonging to numerator function. Further, we can also write the solution set in alternate form as :

x - , - 1 U ( - 1,2 ] U 4,

Determining interval of four quadratic inequalities

Let us take the rational expression of example case and determine intervals of each of four inequalities. The sign diagram as drawn earlier is shown here :

Sign scheme/ diagram

Sign scheme/ diagram

f x < 0 ; x 2,4 f x 0 ; x [ 2,4 ) f x > 0 ; x - , - 1 U - 1,2 U 4, f x 0 ; x - , - 1 U ( - 1,2 ] U 4,

Note that critical point “2” belonging to numerator is included for inequalities which allows equality.

Questions & Answers

Is there any normative that regulates the use of silver nanoparticles?
Damian Reply
what king of growth are you checking .?
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
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
what school?
biomolecules are e building blocks of every organics and inorganic materials.
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
sciencedirect big data base
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.
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
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
characteristics of micro business
for teaching engĺish at school how nano technology help us
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
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
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.
what is the actual application of fullerenes nowadays?
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.
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.
is Bucky paper clear?
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
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.
Do you know which machine is used to that process?
how to fabricate graphene ink ?
for screen printed electrodes ?
What is lattice structure?
s. Reply
of graphene you mean?
or in general
in general
Graphene has a hexagonal structure
On having this app for quite a bit time, Haven't realised there's a chat room in it.
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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What is power set
Satyabrata Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply
Period of sin^6 3x+ cos^6 3x
Sneha Reply

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Source:  OpenStax, Functions. OpenStax CNX. Sep 23, 2008 Download for free at http://cnx.org/content/col10464/1.64
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