3.5 Rational inequality  (Page 3/6)

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$⇒f\left(-2\right)=\frac{{\left(-2\right)}^{2}-\left(-2\right)-2}{{\left(-2\right)}^{2}-3\left(-2\right)-8}=\frac{4+2-1}{4+6-8}=\frac{5}{2}>0$

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

$f\left(x\right)=\frac{{x}^{2}-x-2}{{x}^{2}-3x-8}\ge 0$

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

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

Clearly, solution of given inequality is :

$x\in \left(-\infty ,2\right]U\left(4,\infty \right)-\left\{-1,4\right\}$

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 :

$x\in \left(-\infty ,-1\right)U\left(-1,2\right]U\left(4,\infty \right)$

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 :

$f\left(x\right)<0;\phantom{\rule{1em}{0ex}}x\in \left(2,4\right)$ $f\left(x\right)\le 0;\phantom{\rule{1em}{0ex}}x\in \left[2,4\right)$ $f\left(x\right)>0;\phantom{\rule{1em}{0ex}}x\in \left(-\infty ,-1\right)U\left(-1,2\right)U\left(4,\infty \right)$ $f\left(x\right)\ge 0;\phantom{\rule{1em}{0ex}}x\in \left(-\infty ,-1\right)U\left(-1,2\right]U\left(4,\infty \right)$

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

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
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
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da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
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
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Alexandre
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Rafiq
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Damian
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what is a peer
What is meant by 'nano scale'?
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
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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?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
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Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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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
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What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x