



Consecutive integers have the property that if
$\begin{array}{ccc}\hfill n& =& \text{the smallest integer, then}\hfill \\ \hfill n+1& =& \text{the next integer, and}\hfill \\ \hfill n+2& =& \text{the next integer, and so on.}\hfill \end{array}$
Consecutive odd or even integers have the property that if
$\begin{array}{ccc}\hfill n& =& \text{the smallest integer, then}\hfill \\ \hfill n+2& =& \text{the next odd or even integer (since odd or even numbers differ by 2), and}\hfill \\ \hfill n+4& =& \text{the next odd or even integer, and so on.}\hfill \end{array}$
The sum of three consecutive odd integers is equal to one less than twice the first odd integer. Find the three integers.

$\begin{array}{cccc}Let& \hfill n& =& \text{the first odd integer. Then,}\hfill \\ & \hfill n+2& =& \text{the second odd integer, and}\hfill \\ & \hfill n+4& =& \text{the third odd integer.}\hfill \end{array}$
 Translate the words to mathematical symbols and construct an equation. Read phrases.
$\begin{array}{cc}\text{The sum of:}\hfill & \text{add some numbers}\hfill \\ \text{three consecutive odd integers:}\hfill & n,n+\mathrm{2,}n+4\hfill \\ \text{is equal to:}\hfill & =\hfill \\ \text{one less than:}\hfill & \text{subtract 1 from}\hfill \\ \text{twice the first odd integer:}\hfill & 2n\hfill \end{array}\}n+(n+2)+(n+4)=2n1$

$\begin{array}{cc}n+n+2+n+4=2n1\hfill & \\ 3n+6=2n1\hfill & \text{Subtract}2n\text{from}\mathit{\text{both}}\text{sides.}\hfill \\ 3n+62n=2n12n\hfill \\ n+6=1\hfill & \text{Subtract 6 from}\mathit{\text{both}}\text{sides.}\hfill \\ n+66=16\hfill & \\ n=7\hfill & \text{The first integer is 7.}\hfill \\ n+2=7+2=5\hfill & \text{The second integer is 5.}\hfill \\ n+4=7+4=3\hfill & \text{The third integer is 3.}\hfill \end{array}$
 Check this result.
The sum of the three integers is
$\begin{array}{ccc}\hfill 7+(5)+(3)& =& \text{12}+(3)\hfill \\ & =& \text{15}\hfill \end{array}$
One less than twice the first integer is
$2(7)1=\text{14}1=\text{15}$ . Since these two results are equal, the solution checks.
 The three odd integers are 7, 5, 3.
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Practice set a
The sum of three consecutive even integers is six more than four times the middle integer. Find the integers.
 Let
$x=$ smallest integer.
= next integer.
= largest integer.
 Check:
 The integers are
,
, and
.
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Geometry problems
Sample set b
The perimeter (length around) of a rectangle is 20 meters. If the length is 4 meters longer than the width, find the length and width of the rectangle.
 Let
$x=$ the width of the rectangle. Then,
$x+4=$ the length of the rectangle.
 We can draw a picture.
The length around the rectangle is
$\underset{\text{width}}{\underbrace{x}}+\underset{\text{length}}{\underbrace{\left(x+4\right)}}+\underset{\text{width}}{\underbrace{x}}+\underset{\text{length}}{\underbrace{\left(x+4\right)}}=20$

$\begin{array}{cc}x+x+4+x+x+4=\text{20}\hfill & \\ 4x+8=\text{20}\hfill & \text{Subtract 8 from}\mathit{\text{both}}\text{sides.}\hfill \\ 4x=\text{12}\hfill & \text{Divide}\mathit{\text{both}}\text{sides by 4.}\hfill \\ x=3\hfill & \text{Then,}\hfill \\ x+4=3+4=7\hfill & \end{array}$
 Check:
 The length of the rectangle is 7 meters.
The width of the rectangle is 3 meters.
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Practice set b
The perimeter of a triangle is 16 inches. The second leg is 2 inches longer than the first leg, and the third leg is 5 inches longer than the first leg. Find the length of each leg.
 Let
$x=$ length of the first leg.
= length of the second leg.
= length of the third leg.
 We can draw a picture.
 Check:
 The lengths of the legs are
,
, and
.
3 inches, 5 inches, and 8 inches
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Exercises
For the following 17 problems, find each solution using the fivestep method.
Questions & Answers
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
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
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
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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 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?
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
biomolecules are e building blocks of every organics and inorganic materials.
Joe
7hours 36 min  4hours 50 min
Source:
OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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