# 3.5 Rational inequality  (Page 6/6)

 Page 6 / 6

$x={y}^{n}$

Interchangeably, we write :

$y={x}^{\frac{1}{n}}={}_{\sqrt{}}^{n}\left(x\right)$

If n is even integer, then x can not be negative. For n=2, we drop “n” from the notation and we write,

$y=\sqrt{x}$

We extend this concept to function in which number “x” is substituted by any valid expression (algebraic, trigonometric, logaritmic etc). Some examples are :

$y=\sqrt{{x}^{2}+3x-5}$ $y=\sqrt{\mathrm{log}{}_{e}\left({x}^{2}+3x-5\right)}$

We shall also include study of radical function which is part of rational form like :

$y=\sqrt{\left(\frac{1}{{x}_{2}+3x-5}\right)}$

$y=\sqrt{\left\{\frac{\left(x-1\right)\left(x+3\right)}{{x}_{2}+3x-5}\right\}}$

Analysis of root function is same as analysis of inequality of function. Because, radical function ultimately results in inequality. We make use of the fact that expression within the radical sign is non-negative. Here, we denote a radical function as :

$f\left(x\right)=\sqrt{g\left(x\right)}$

As the expression under is non-negative,

$⇒g\left(x\right)\ge 0$

When radical function is part of a function defined in rational form, the radical function should not be zero. Let us consider a function as :

$f\left(x\right)=\frac{1}{\sqrt{g\left(x\right)}}$

As the radical is denominator of the rational expression, expression under radical sign is positive,

$g\left(x\right)>0$

We have already worked with inequalities involving polynomial and rational functions. We shall restrict ourselves to few illustrations here.

Problem : Find domain of the function :

$f\left(x\right)=\sqrt{\left\{1-\sqrt{\left(1-{x}^{2}\right)}\right\}}$

$⇒1-\sqrt{1-{x}^{2}}\ge 0$ $⇒\sqrt{1-{x}^{2}}\le 1$

The term on each side of inequality is a positive quantity. Squaring each side does not change inequality,

$⇒1-{x}^{2}\le 1$ $⇒{x}^{2}\ge 0$

This quadratic inequality is true for all real x. Now, for inner radical

$⇒1-{x}^{2}\ge 0$ $⇒\left(1+x\right)\left(1-x\right)\ge 0$

We multiply by -1 to change the sign of x in 1-x,

$⇒\left(x+1\right)\left(x-1\right)\le 0$

Using sign rule :

$⇒x\in \left[-1,1\right]$

Since, conditions corresponding to two radicals need to be fulfilled simultaneously, the domain of the given function is intersection of outer and inner radicals.

$\text{Domain}=\left[-1,1\right]$

Problem : Find the domain of the function given by :

$f\left(x\right)=\sqrt{\left({x}^{14}-{x}^{11}+{x}^{6}-{x}^{3}+{x}^{2}+1\right)}$

Solution : Clearly, function is real for values of “x” for which expression within square root is a non negative number. We note that independent variable is raised to positive integers. The nature of each monomial depends on the value of x and nature of power. If x≥1, then monomial evaluates to higher value for higher power. If x lies between 0 and 1, then monomial evaluates to lower value for higher power. Further, a negative x yields negative value when raised to odd power and positive value when raised to even power. We shall use these properties to evaluate the expression for three different intervals of x.

$⇒{x}^{14}-{x}^{11}+{x}^{6}-{x}^{3}+{x}^{2}+1\ge 0$

We consider different intervals of values of expression for different values of “x”, which cover the complete interval of real numbers.

1: $x\ge 1$

In this case, ${x}^{a}>{x}^{b},\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}a>b$ . Evaluating in groups,

$\left({x}^{14}-{x}^{11}\right)+\left({x}^{6}-{x}^{3}\right)+\left({x}^{2}+1\right)>0$

2: $0\le x<1$

In this case, ${x}^{a}<{x}^{b},\phantom{\rule{1em}{0ex}}\text{if}\phantom{\rule{1em}{0ex}}a>b$ . Rearranging in groups,

${x}^{14}-\left\{\left({x}^{11}-{x}^{6}\right)+\left({x}^{3}-{x}^{2}\right)\right\}+1$

Here, $\left\{\left({x}^{11}-{x}^{6}\right)+\left({x}^{3}-{x}^{2}\right)\right\}$ is negative. Hence total expression is positive,

$⇒{x}^{14}-\left\{\left({x}^{11}-{x}^{6}\right)+\left({x}^{3}-{x}^{2}\right)\right\}+1>0$

3: $x<0$

Rearranging in groups,

$\left({x}^{14}-{x}^{11}\right)+\left({x}^{6}-{x}^{3}\right)+\left({x}^{2}+1\right)$

Here, ${x}^{14,}{x}^{6}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{x}^{2}$ are positive and ${x}^{11}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{x}^{3}$ is negative. Hence, total expression is positive,

$⇒\left({x}^{14}-{x}^{11}\right)+\left({x}^{6}-{x}^{3}\right)+\left({x}^{2}+1\right)>0$

We see that expression is positive for all values of “x”. Hence, domain of the function is :

$\text{Domain}=R=\left(-\infty ,\infty \right)$

## Exercise

Find solution of the rational inequality given by :

$\frac{\left(x+1\right)\left(x+5\right)}{\left(x-3\right)}\ge 0$

Hint : Critical points are -5,-1 and 3. We need to exclude end corresponding to x=3 as denominator turns zero for this value.

$\left[-5,-1\right]\phantom{\rule{1em}{0ex}}\cup \phantom{\rule{1em}{0ex}}\left(3,\infty \right)$

Find solution of the rational inequality given by :

$\frac{8{x}^{2}+16x-51}{\left(2x-3\right)\left(x+4\right)}>0$

Hint : Critical points are -4,-3,3/2,5/2.

$⇒x\in \left(-\infty ,-4\right)\cup \left(-3,3/2\right)\cup \left(5/2,\infty \right)$

Find solution of the rational inequality given by :

$\frac{{x}^{2}+4x+3}{{x}^{3}-6{x}^{2}+11x-6}>0$

Hint : Factorize denominator as ${x}^{3}-6{x}^{2}+11x-6=\left(x-1\right)\left(x-2\right)\left(x-3\right)$ . Critical points are -3,-1,1,2,3.

$⇒x\in \left(-3,-1\right)\cup \left(1,2\right)\cup \left(3,\infty \right)$

Find solution of the rational inequality given by :

$\frac{\left(2x+1\right){\left(x-1\right)}^{2}}{\left({x}^{3}-3{x}^{2}+2x\right)}\ge 0$

Hint : Factorize denominator as

$⇒\frac{\left(2x+1\right){\left(x-1\right)}^{2}}{\left({x}^{3}-3{x}^{2}+2x\right)}=\frac{\left(2x+1\right){\left(x-1\right)}^{2}}{x\left(x-1\right)\left(x-2\right)}$

Critical points are -1/2,1,1,0,1 and 2. We see that "1" is repeated odd times. Hence, we continue to assign alternating signs in accordance with wavy curve method. The solution of x for the inequality is :

$-\infty

where we get a research paper on Nano chemistry....?
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
learn
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
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
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
nanocopper obvius
Alexandre
what is the stm
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?
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
what is Nano technology ?
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?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
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
Privacy Information Security Software Version 1.1a
Good
What is power set
Period of sin^6 3x+ cos^6 3x
Period of sin^6 3x+ cos^6 3x By Rhodes By OpenStax By Candice Butts By Olivia D'Ambrogio By OpenStax By By Dan Ariely By OpenStax By Brooke Delaney By Steve Gibbs