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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

what is the stm
Brian Reply
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
scanning tunneling microscope
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Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
what is differents between GO and RGO?
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
what is Nano technology ?
Bob Reply
write examples of Nano molecule?
The nanotechnology is as new science, to scale nanometric
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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
How can I make nanorobot?
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
how can I make nanorobot?
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.
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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