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Consecutive integers have the property that if

n = the smallest integer, then n + 1 = the next integer, and n + 2 = the next integer, and so on.

Consecutive odd or even integers have the property that if

n = the smallest integer, then n + 2 = the next odd or even integer (since odd or even numbers differ by 2), and n + 4 = the next odd or even integer, and so on.

The sum of three consecutive odd integers is equal to one less than twice the first odd integer. Find the three integers.

  1. Let n = the first odd integer. Then, n + 2 = the second odd integer, and n + 4 = the third odd integer.
  2. Translate the words to mathematical symbols and construct an equation. Read phrases.

    The sum of: add some numbers three consecutive odd integers: n , n + 2, n + 4 is equal to: = one less than: subtract 1 from twice the first odd integer: 2 n } n + ( n + 2 ) + ( n + 4 ) = 2 n 1 size 12{ left none matrix { "The sum of:" {} # "add some numbers" {} ##"three consecutive odd integers:" {} # n,n+2,n+4 {} ## "is equal to:" {} # ={} {} ##"one less than:" {} # "subtract 1 from" {} ## "twice the first odd integer:" {} # 2n{}} right rbrace n+ \( n+2 \) + \( n+4 \) =2n - 1} {}

  3. n + n + 2 + n + 4 = 2 n 1 size 12{n+n+2+n+4=2n - 1} {} 3 n + 6 = 2 n 1 size 12{3n+6=2n - 1} {} Subtract 2 n size 12{2n} {} from  both  sides. 3 n + 6 2 n = 2 n 1 2 n size 12{3n+6 - 2n=2n - 1 - 2n} {} n + 6 = 1 size 12{n+6= - 1} {} Subtract 6 from  both  sides. n + 6 6 = 1 6 size 12{n+6 - 6= - 1 - 6} {} n = 7 size 12{n= - 7} {} The first integer is -7. n + 2 = 7 + 2 = 5 size 12{n+2= - 7+2= - 5} {} The second integer is -5. n + 4 = 7 + 4 = 3 size 12{n+4= - 7+4= - 3} {} The third integer is -3.

  4. Check this result.
    The sum of the three integers is

    7 + ( 5 ) + ( 3 ) = 12 + ( 3 ) = 15

    One less than twice the first integer is 2 ( 7 ) 1 = 14 1 = 15 size 12{2 \( - 7 \) - 1= - "14" - 1= - "15"} {} . Since these two results are equal, the solution checks.

  5. The three odd integers are -7, -5, -3.
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Practice set a

When three times a number is decreased by 5, the result is -23. Find the number.

  1. Let x =
  2. Check:
  3. The number is .

-6

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When five times a number is increased by 7, the result is five less than seven times the number. Find the number.

  1. Let n =
  2. Check:
  3. The number is .

6

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Two consecutive numbers add to 35. Find the numbers.

  1. Check:
  2. The numbers are and .

17 and 18

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The sum of three consecutive even integers is six more than four times the middle integer. Find the integers.

  1. Let x = smallest integer.
    = next integer.
    = largest integer.
  2. Check:
  3. The integers are , , and .

-8, -6, and -4

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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.

  1. Let x = size 12{x={}} {} the width of the rectangle. Then,
    x + 4 = size 12{x+4={}} {} the length of the rectangle.
  2. We can draw a picture.

    A rectangle with base length the quantity x + 4 and height x.

    The length around the rectangle is
    x width + x + 4 length + x width + x + 4 length = 20

  3. x + x + 4 + x + x + 4 = 20 size 12{x+x+4+x+x+4="20"} {} 4 x + 8 = 20 size 12{4x+8="20"} {} Subtract 8 from  both  sides. 4 x = 12 size 12{4x="12"} {} Divide  both  sides by 4. x = 3 size 12{x=3} {} Then, x + 4 = 3 + 4 = 7 size 12{x+4=3+4=7} {}

  4. Check:

    A rectangle with base length 7 and height 3.

  5. 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.

  1. Let x = size 12{x={}} {} length of the first leg.
    = length of the second leg.
    = length of the third leg.
  2. We can draw a picture.
  3. Check:
  4. 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 five-step method.

What number decreased by nine is fifteen?

  1. Let n = size 12{n={}} {} the number.
  2. Check:
  3. The number is .

24

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What number increased by twelve is twenty?

  1. Let n = size 12{n={}} {} the number.
  2. Check:
  3. The number is .
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If five more than three times a number is thirty-two, what is the number?

  1. Let x = size 12{x={}} {} the number.
  2. Check:
  3. The number is .

9

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Questions & Answers

where we get a research paper on Nano chemistry....?
Maira Reply
what are the products of Nano chemistry?
Maira Reply
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
Google
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
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
Jyoti Reply
I only see partial conversation and what's the question here!
Crow Reply
what about nanotechnology for water purification
RAW Reply
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
Brian Reply
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?
LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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 ?
Bob Reply
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?
Damian Reply
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
Stoney Reply
why we need to study biomolecules, molecular biology in nanotechnology?
Adin Reply
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
Adin
why?
Adin
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
7hours 36 min - 4hours 50 min
Tanis Reply

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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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