Page 1 / 1

## Introduction

Now that you know how to solve quadratic equations, you are ready to learn how to solve quadratic inequalities.

A quadratic inequality is an inequality of the form

$\begin{array}{c}\hfill a{x}^{2}+bx+c>0\\ \hfill a{x}^{2}+bx+c\ge 0\\ \hfill a{x}^{2}+bx+c<0\\ \hfill a{x}^{2}+bx+c\le 0\end{array}$

Solving a quadratic inequality corresponds to working out in what region the graph of a quadratic function lies above or below the $x$ -axis.

Solve the inequality $4{x}^{2}-4x+1\le 0$ and interpret the solution graphically.

1. Let $f\left(x\right)=4{x}^{2}-4x+1$ . Factorising this quadratic function gives $f\left(x\right)={\left(2x-1\right)}^{2}$ .

2. ${\left(2x-1\right)}^{2}\le 0$
3. $f\left(x\right)=0$ only when $x=\frac{1}{2}$ .

4. This means that the graph of $f\left(x\right)=4{x}^{2}-4x+1$ touches the $x$ -axis at $x=\frac{1}{2}$ , but there are no regions where the graph is below the $x$ -axis.

Find all the solutions to the inequality ${x}^{2}-5x+6\ge 0$ .

1. The factors of ${x}^{2}-5x+6$ are $\left(x-3\right)\left(x-2\right)$ .

2. $\begin{array}{ccc}\hfill {x}^{2}-5x+6& \ge & 0\hfill \\ \hfill \left(x-3\right)\left(x-2\right)& \ge & 0\hfill \end{array}$
3. We need to figure out which values of $x$ satisfy the inequality. From the answers we have five regions to consider.

4. Let $f\left(x\right)={x}^{2}-5x+6$ . For each region, choose any point in the region and evaluate the function.

 $f\left(x\right)$ sign of $f\left(x\right)$ Region A $x<2$ $f\left(1\right)=2$ + Region B $x=2$ $f\left(2\right)=0$ + Region C $2 $f\left(2,5\right)=-2,5$ - Region D $x=3$ $f\left(3\right)=0$ + Region E $x>3$ $f\left(4\right)=2$ +

We see that the function is positive for $x\le 2$ and $x\ge 3$ .

5. We see that ${x}^{2}-5x+6\ge 0$ is true for $x\le 2$ and $x\ge 3$ .

Solve the quadratic inequality $-{x}^{2}-3x+5>0$ .

1. Let $f\left(x\right)=-{x}^{2}-3x+5$ . $f\left(x\right)$ cannot be factorised so, use the quadratic formula to determine the roots of $f\left(x\right)$ . The $x$ -intercepts are solutions to the quadratic equation

$\begin{array}{ccc}\hfill -{x}^{2}-3x+5& =& 0\hfill \\ \hfill {x}^{2}+3x-5& =& 0\hfill \\ \hfill \therefore x& =& \frac{-3±\sqrt{{\left(3\right)}^{2}-4\left(1\right)\left(-5\right)}}{2\left(1\right)}\hfill \\ & =& \frac{-3±\sqrt{29}}{2}\hfill \\ \hfill {x}_{1}& =& \frac{-3-\sqrt{29}}{2}\hfill \\ \hfill {x}_{2}& =& \frac{-3+\sqrt{29}}{2}\hfill \end{array}$
2. We need to figure out which values of $x$ satisfy the inequality. From the answers we have five regions to consider.

3. We can use another method to determine the sign of the function over different regions, by drawing a rough sketch of the graph of the function. We know that the roots of the function correspond to the $x$ -intercepts of the graph. Let $g\left(x\right)=-{x}^{2}-3x+5$ . We can see that this is a parabola with a maximum turning point that intersects the $x$ -axis at ${x}_{1}$ and ${x}_{2}$ .

It is clear that $g\left(x\right)>0$ for ${x}_{1}

4. $-{x}^{2}-3x+5>0$ for ${x}_{1}

When working with an inequality where the variable is in the denominator, a different approach is needed.

Solve $\frac{2}{x+3}\le \frac{1}{x-3}$

1. $\frac{2}{x+3}-\frac{1}{x-3}\le 0$
2. $\begin{array}{c}\hfill \frac{2\left(x-3\right)-\left(x+3\right)}{\left(x+3\right)\left(x-3\right)}\le 0\\ \hfill \frac{x-9}{\left(x+3\right)\left(x-3\right)}\le 0\end{array}$
3. We see that the expression is negative for $x<-3$ or $3 .

4. $x<-3\phantom{\rule{1.em}{0ex}}or\phantom{\rule{1.em}{0ex}}3

## End of chapter exercises

Solve the following inequalities and show your answer on a number line.

1. Solve: ${x}^{2}-x<12$ .
2. Solve: $3{x}^{2}>-x+4$
3. Solve: ${y}^{2}<-y-2$
4. Solve: $-{t}^{2}+2t>-3$
5. Solve: ${s}^{2}-4s>-6$
6. Solve: $0\ge 7{x}^{2}-x+8$
7. Solve: $0\ge -4{x}^{2}-x$
8. Solve: $0\ge 6{x}^{2}$
9. Solve: $2{x}^{2}+x+6\le 0$
10. Solve for $x$ if: $\frac{x}{x-3}<2$ and $x\ne 3$ .
11. Solve for $x$ if: $\frac{4}{x-3}\le 1$ .
12. Solve for $x$ if: $\frac{4}{{\left(x-3\right)}^{2}}<1$ .
13. Solve for $x$ : $\frac{2x-2}{x-3}>3$
14. Solve for $x$ : $\frac{-3}{\left(x-3\right)\left(x+1\right)}<0$
15. Solve: ${\left(2x-3\right)}^{2}<4$
16. Solve: $2x\le \frac{15-x}{x}$
17. Solve for $x$ :     $\frac{{x}^{2}+3}{3x-2}\le 0$
18. Solve: $x-2\ge \frac{3}{x}$
19. Solve for $x$ : $\frac{{x}^{2}+3x-4}{5+{x}^{4}}\le 0$
20. Determine all real solutions: $\frac{x-2}{3-x}\ge 1$

are nano particles real
yeah
Joseph
Hello, if I study Physics teacher in bachelor, can I study Nanotechnology in master?
no can't
Lohitha
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
learn
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
has a lot of application modern world
Kamaluddeen
yes
narayan
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
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
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
what is hormones?
Wellington
Other chapter Q/A we can ask