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Now let’s see what happens when the signs are different.

Model: −5 + 3 .

Solution

Interpret the expression. −5 + 3 means the sum of −5 and 3 .
Model the first number. Start with 5 negatives. .
Model the second number. Add 3 positives. .
Remove any neutral pairs. .
Count the result. .
The sum of −5 and 3 is −2. −5 + 3 = −2

Notice that there were more negatives than positives, so the result is negative.

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Model the expression, and then simplify:

2 + ( −4 )


This figure shows two rows of counter circles. The first row has 2 dark pink circles, representing negative counters. The second row has 4 light pink circles, representing positive counters.
2

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Model the expression, and then simplify:

2 + ( −5 )


This figure shows two rows of counter circles. The first row has 2 light pink circles, representing positive counters. The second row has 5 dark pink circles, representing negative counters.
−3

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Model: 5 + ( −3 ) .

Solution

Interpret the expression. 5 + ( −3 ) means the sum of 5 and −3 .
Model the first number. Start with 5 positives. .
Model the second number. Add 3 negatives. .
Remove any neutral pairs. .
Count the result. .
The sum of 5 and −3 is 2. 5 + ( −3 ) = 2
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Model the expression, and then simplify:

( −2 ) + 4


This figure shows two rows of counter circles. The first row has 2 light pink circles, representing positive counters. The second row has 4 dark pink circles, representing negative counters.
−2

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Model the expression:

( −2 ) + 5


This figure shows two rows of counters. The first row shows 2 dark pink circles, representing negative counters. The second row has 5 light pink circles, representing negative counters. The neutral pairs of one positive and one negative counter are circled leaving three positive counters.
3

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Modeling addition of positive and negative integers

Model each addition.

  1. ⓐ 4 + 2
  2. ⓑ −3 + 6
  3. ⓒ 4 + (−5)
  4. ⓓ -2 + (−3)

4 + 2
Start with 4 positives. .
Add two positives. .
How many do you have? 6 4 + 2 = 6
3 + 6
Start with 3 negatives. .
Add 6 positives. .
Remove neutral pairs. .
How many are left? .
3 −3 + 6 = 3
4 + ( −5 )
Start with 4 negatives. .
Add 5 negatives. .
Remove neutral pairs. .
How many are left? .
−1 4 + ( −5 ) = −1
−2 + ( −3 )
Start with 2 negatives. .
Add 3 negatives. .
How many do you have? −5 −2 + ( −3 ) = −5

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Model each addition.

  1. ⓐ 3 + 4
  2. ⓑ −1 + 4
  3. ⓒ 4 + (−6)
  4. ⓓ−2 + (−2)


  1. .

  2. .

  3. .

  4. .

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  1. ⓐ 5 + 1
  2. ⓑ −3 + 7
  3. ⓒ 2 + (−8)
  4. ⓓ −3 + (−4)


  1. .

  2. .

  3. .

  4. .

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Simplify expressions with integers

Now that you have modeled adding small positive and negative integers, you can visualize the model in your mind to simplify expressions    with any integers.

For example, if you want to add 37 + ( −53 ) , you don’t have to count out 37 blue counters and 53 red counters.

Picture 37 blue counters with 53 red counters lined up underneath. Since there would be more negative counters than positive counters, the sum would be negative. Because 53 −37 = 16 , there are 16 more negative counters.

37 + ( −53 ) = −16

Let’s try another one. We’ll add −74 + ( −27 ) . Imagine 74 red counters and 27 more red counters, so we have 101 red counters all together. This means the sum is −101.

−74 + ( −27 ) = −101
Addition of positive and negative integers
5 + 3 −5 + ( −3 )
both positive, sum positive both negative, sum negative
When the signs are the same, the counters would be all the same color, so add them.
−5 + 3 5 + ( −3 )
different signs, more negatives different signs, more positives
Sum negative sum positive
When the signs are different, some counters would make neutral pairs; subtract to see how many are left.

Simplify:

  1. 19 + ( −47 )
  2. −32 + 40

Solution

ⓐ Since the signs are different, we subtract 19 from 47 . The answer will be negative because there are more negatives than positives.

19 + ( −47 ) −28

ⓑ The signs are different so we subtract 32 from 40 . The answer will be positive because there are more positives than negatives

−32 + 40 8
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Simplify each expression:

  1. 15 + ( −32 )
  2. −19 + 76
  1. ⓐ −17
  2. ⓑ 57
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Simplify each expression:

  1. −55 + 9
  2. 43 + ( −17 )

  1. ⓐ −46
  2. ⓑ 26

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Simplify: −14 + ( −36 ) .

Solution

Since the signs are the same, we add. The answer will be negative because there are only negatives.

−14 + ( −36 ) −50
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Simplify the expression:

−31 + ( −19 )

−50

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Simplify the expression:

−42 + ( −28 )

−70

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Questions & Answers

anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
Introduction about quantum dots in nanotechnology
Praveena Reply
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
Damian Reply
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
Akash Reply
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
s. Reply
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
Devang Reply
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Abhijith Reply
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
s. Reply
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
SUYASH Reply
for screen printed electrodes ?
SUYASH
What is lattice structure?
s. Reply
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
Sanket Reply
what's the easiest and fastest way to the synthesize AgNP?
Damian Reply
China
Cied
types of nano material
abeetha Reply
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
how did you get the value of 2000N.What calculations are needed to arrive at it
Smarajit Reply
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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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