# 11.6 Applications ii: solving problems  (Page 2/3)

 Page 2 / 3

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.

1. $\begin{array}{cccc}Let& \hfill n& =& \text{the first odd integer. Then,}\hfill \\ & \hfill n+2& =& \text{the second odd integer, and}\hfill \\ & \hfill n+4& =& \text{the third odd integer.}\hfill \end{array}$
2. Translate the words to mathematical symbols and construct an equation. Read phrases.

$\begin{array}{cc}\text{The sum of:}\hfill & \text{add some numbers}\hfill \\ \text{three consecutive odd integers:}\hfill & n,n+2,n+4\hfill \\ \text{is equal to:}\hfill & =\hfill \\ \text{one less than:}\hfill & \text{subtract 1 from}\hfill \\ \text{twice the first odd integer:}\hfill & 2n\hfill \end{array}\right\}n+\left(n+2\right)+\left(n+4\right)=2n-1$

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

$\begin{array}{ccc}\hfill -7+\left(-5\right)+\left(-3\right)& =& -\text{12}+\left(-3\right)\hfill \\ & =& -\text{15}\hfill \end{array}$

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

4. The three odd integers are -7, -5, -3.

## 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

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

Two consecutive numbers add to 35. Find the numbers.

1. Check:
2. The numbers are and .

17 and 18

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

## 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 $x=$ the width of the rectangle. Then,
$x+4=$ the length of the rectangle.
2. We can draw a picture. The length around the rectangle is
$\underset{\text{width}}{\underbrace{x}}+\underset{\text{length}}{\underbrace{\left(x+4\right)}}+\underset{\text{width}}{\underbrace{x}}+\underset{\text{length}}{\underbrace{\left(x+4\right)}}=20$

3. Check:  4. The length of the rectangle is 7 meters.
The width of the rectangle is 3 meters.

## 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 $x=$ 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

## Exercises

For the following 17 problems, find each solution using the five-step method.

What number decreased by nine is fifteen?

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

24

What number increased by twelve is twenty?

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

If five more than three times a number is thirty-two, what is the number?

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

9

#### Questions & Answers

how can chip be made from sand
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Almas
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Missy Reply
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Joseph
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no can't
Lohitha
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William
currently
William
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Maira Reply
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
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Maira Reply
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learn
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Google
da
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Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
Hafiz Reply
revolt
da
Application of nanotechnology in medicine
has a lot of application modern world
Kamaluddeen
yes
narayan
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Jyoti Reply
ya I also want to know the raman spectra
Bhagvanji
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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
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Alexandre
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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
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LITNING Reply
what is a peer
LITNING Reply
What is meant by 'nano scale'?
LITNING Reply
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LITNING
scanning tunneling microscope
Sahil
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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
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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
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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
In the number 779,844,205 how many ten millions are there?
TELLY Reply
From 1973 to 1979, in the United States, there was an increase of 166.6% of Ph.D. social scien­tists to 52,000. How many were there in 1973?
Khizar Reply
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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