3.5 Rational inequality  (Page 5/6)

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Problem : Find solution of :

$\frac{2}{1+x}+\frac{3}{1-x}<1$

Solution : Rearranging, we have :

$⇒\frac{2-2x+3+3x}{\left(1+x\right)\left(1-x\right)}-1<0$ $⇒\frac{5+x-\left(1-{x}^{2}\right)}{\left(1+x\right)\left(1-x\right)}<0$ $⇒\frac{{x}^{2}+x+4}{\left(1+x\right)\left(1-x\right)}<0$

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 :

$⇒\frac{1}{\left(1+x\right)\left(1-x\right)}>0$

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

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

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, ${x}^{3}-6{x}^{2}+11x-6=\left(x-1\right)\left({x}^{2}-5x+6\right)=\left(x-1\right)\left(x-2\right)\left(x-3\right)$

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

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

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

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

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.

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

$-1/2\le x<0\phantom{\rule{1em}{0ex}}\cup \phantom{\rule{1em}{0ex}}2

$\left[-1/2,0\right)\phantom{\rule{1em}{0ex}}\cup \phantom{\rule{1em}{0ex}}\left(2,\infty \right)$

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, $f\left(x\right)=2{x}^{2}+x-1<0$ . 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 :

Solution of x satisfying inequality is :

$x\in \left(-2,\frac{3}{2}\right)$

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

The term radical is name given to square root sign (√). A radical number is ${n}^{th}$ root of a real number. If y is ${n}^{th}$ root of x, then :

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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
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Preparation and Applications of Nanomaterial for Drug Delivery
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I only see partial conversation and what's the question here!
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Professor
I think
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if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
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analytical skills graphene is prepared to kill any type viruses .
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write examples of Nano molecule?
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The nanotechnology is as new science, to scale nanometric
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nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
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yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
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biomolecules are e building blocks of every organics and inorganic materials.
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how did you get the value of 2000N.What calculations are needed to arrive at it
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