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Problem : Find solution of :

2 1 + x + 3 1 x < 1

Solution : Rearranging, we have :

2 2 x + 3 + 3 x 1 + x 1 x 1 < 0 5 + x 1 x 2 1 + x 1 x < 0 x 2 + x + 4 1 + x 1 x < 0

Now, polynomial in the numerator i.e. x 2 + x + 4 is positive for all real x as D<0 and a>0. Thus, dividing either side of the inequality by this polynomial does not change inequality. Now, we need to change the sign of x in one of the linear factors of the denominator positive in accordance with sign rule. This is required to be done in the factor (1-x). For this, we multiply each side of inequality by -1. This change in sign accompanies change in inequality as well :

1 1 + x 1 x > 0

Critical points are -1 and 1. Hence, solution of the inequality in x is :

Sign diagram

Sign of function alternates.

x - , - 1 1,

Rational inequality with repeated linear factors

We have already discussed rational polynomial with repeated factors. We need to count repeated factors which appear in both numerator and denominator. If the linear factors are repeated even times, then we do not need to change sign about critical point corresponding to repeated linear factor.

Note : While working with rational function having repeated factors, we need to factorize higher order polynomial like cubic polynomial. In such situation, we can employ a short cut. We guess one real root of the cubic polynomial. We may check corresponding equation with values such as 1,2, -1 or -2 etc and see whether cubic expression becomes zero or not for that value. If one of the roots is known, then cubic expression is f(x) = (x-a) g(x), where "a" is the guessed root and g(x) is a quadratic expression. We can then find other two roots anlayzing quadratic expression. For example, x 3 6 x 2 + 11 x 6 = x 1 ( x 2 - 5 x + 6 ) = x 1 x 2 x 3

Problem : Find interval of x satisfying the inequality given by :

2 x + 1 x 1 x 3 3 x 2 + 2 x 0

Solution : We factorize each of the polynomials in numerator and denominator :

2 x + 1 x 1 x 3 3 x 2 + 2 x = 2 x + 1 x 1 x x 1 x 2

It is important that we do not cancel common factors or terms. Here, critical points are -1/2,1,0,1 and 2. The critical point "1" is repeated even times. Hence, we do not change sign about "1" while drawing sign scheme.

Sign diagram

Sign of function alternates.

While writing interval, we drop equality sign for critical points, which corresponds to denominator.

- 1 / 2 x < 0 2 < x <

[ - 1 / 2 , 0 ) ( 2 , )

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

Polynomial inequality

We can treat polynomial inequality as rational inequality, because a polynomial function is a rational function with denominator as 1. Logically, sign method used for rational function should also hold for polynomial function. Let us consider a simple polynomial inequality, f x = 2 x 2 + x - 1 < 0 . Here, function is product of two linear factors (2x-3)(x+2). Clearly, x=3/2 and x=-2 are the critical points. The sign scheme of the function is shown in the figure :

Sign diagram

Sign of function alternates.

Solution of x satisfying inequality is :

x - 2 , 3 2

It is evident that this method is easier and mechanical in approach.

Radical function

The term radical is name given to square root sign (√). A radical number is n t h root of a real number. If y is n t h root of x, then :

Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
Google
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
what is the stm
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
what is Nano technology ?
Bob Reply
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?
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
Privacy Information Security Software Version 1.1a
Good
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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