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Inclusion and exclusion of critical points

Based on the discussion above, we summarize inclusion or exclusion of critical points here :

1: Question of inclusion of critical points arises when inequality involved is not strict.

2: Critical points belonging to numerator are included in solution set.

3: Critical points belonging to denominator are excluded from solution set.

3: Critical points belonging to both numerator and denominator are excluded from solution set.

Solution of rational inequalities using wavy curve method

Wavy curve method is a modified sign diagram method. This method has the advantage that we do not need to test sign of interval as required in earlier case. The steps involved are :

1: Factorize numerator and denominator into linear factors.

2: Make coefficients of x positive in all linear factors. This step may require to change sign of “x” in the linear factor by multiplying inequality with -1. Note that this multiplication will change the inequality sign as well. For example, “less than” will become “greater than” etc.

3: Equate each linear factor to zero and find values of x in each case. The values are called critical points.

4: Identify distinct critical points on real number line. The “n” numbers of distinct critical points divide real number lines in (n+1) sub-intervals.

5: The sign of rational function in the right most interval is positive. Alternate sign in adjoining intervals on the left.

5: 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.

We need to exclude exception points i.e. critical points of denominator from solution set. Further, it is important to understand that signs of intervals as determined using this method are not the signs of function – rather signs of modified function in which sign of “x” has changed. However, if we are not required to change the sign of “x” i.e. to modify the function, then signs of intervals are also signs of function. We shall though keep this difference in mind, but we shall refer signs of intervals as sign scheme or diagram in this case also.

Problem : Apply wavy curve method to find the interval of x for the inequality given :

x 1 x 0

Solution : We change the sign of "x" in the denominator to positive by multiplying both sides of inequality with -1. Note that this changes the inequality sign as well.

x x 1 0

Here, critical points are :

x = 0 , 1

The critical points are marked on the real number line. Starting with positive sign in the right most interval, we denote signs of adjacent intervals by alternating sign.

Sign diagram

Sign of function alternates.

Thus, interval of x as solution of inequality is :

0 x < 1

We do not include "1" as it reduces denominator to zero.

Problem : Find solution of the rational inequality given by :

3 x 2 + 6 x 15 2 x - 1 x + 3 1

Solution : We first convert the given inequality to standard form f(x) ≥ 0.

3 x 2 + 6 x 15 2 x - 1 x + 3 - 1 0

3 x 2 + 6 x 15 2 x - 1 x + 3 2 x - 1 x + 3 0

3 x 2 + 6 x 15 2 x 2 + 5 x 3 2 x - 1 x + 3 0

x 2 + x 12 2 x - 1 x + 3 0 x 2 + 4 x 3 x 12 2 x - 1 x + 3 0 x 3 x + 4 2 x - 1 x + 3 0

Critical points are -4, -3, 1/2, 3. Corresponding sign diagram is :

Sign diagram

Sign of function alternates.

The solution of inequality is :

x - , - 4 ] - 3,1 / 2 [ 3,

We do not include "-3" and "1" as they reduce denominator to zero.

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
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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
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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
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how did you get the value of 2000N.What calculations are needed to arrive at it
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What is power set
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Period of sin^6 3x+ cos^6 3x
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Period of sin^6 3x+ cos^6 3x
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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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