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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The basic operations with real numbers are presented in this chapter. The concept of absolute value is discussed both geometrically and symbolically. The geometric presentation offers a visual understanding of the meaning of |x|. The symbolic presentation includes a literal explanation of how to use the definition. Negative exponents are developed, using reciprocals and the rules of exponents the student has already learned. Scientific notation is also included, using unique and real-life examples.Objectives of this module: understand the geometric and algebraic definitions of absolute value.

Overview

  • Geometric Definition of Absolute Value
  • Algebraic Definition of Absolute Value

Geometric definition of absolute value

Absolute value—geometric approach

The absolute value of a number a , denoted | a | , is the distance from a to 0 on the number line.

Absolute value speaks to the question of "how far," and not "which way." The phrase how far implies length, and length is always a nonnegative (zero or positive) quantity. Thus, the absolute value of a number is a nonnegative number. This is shown in the following examples:

| 5 | = 5 .
The quantity on the left side of the equal sign is read as "negative the absolute value of 5." The absolute value of 5 is 5. Hence, negative the absolute value of 5 is 5 .

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| 3 | = 3 .
The quantity on the left side of the equal sign is read as "negative the absolute value of 3 ." The absolute value of 3 is 3. Hence, negative the absolute value of 3 is ( 3 ) = 3 .

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Algebraic definition of absolute value

The problems in the first example may help to suggest the following algebraic definition of absolute value. The definition is interpreted below. Examples follow the interpretation.

Absolute value—algebraic approach

The absolute value of a number a is

| a | = { a if a 0 a if a < 0

The algebraic definition takes into account the fact that the number a could be either positive or zero ( 0 ) or negative ( < 0 ) .

  1. If the number a is positive or zero ( 0 ) , the first part of the definition applies. The first part of the definition tells us that if the number enclosed in the absolute bars is a nonnegative number, the absolute value of the number is the number itself.
  2. If the number a is negative ( < 0 ) , the second part of the definition applies. The second part of the definition tells us that if the number enclosed within the absolute value bars is a negative number, the absolute value of the number is the opposite of the number. The opposite of a negative number is a positive number.

Sample set a

Use the algebraic definition of absolute value to find the following values.

| 8 | .
The number enclosed within the absolute value bars is a nonnegative number so the first part of the definition applies. This part says that the absolute value of 8 is 8 itself.

| 8 | = 8

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| 3 | .
The number enclosed within absolute value bars is a negative number so the second part of the definition applies. This part says that the absolute value of 3 is the opposite of 3 , which is ( 3 ) . By the double-negative property, ( 3 ) = 3 .

| 3 | = 3

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Practice set a

Use the algebraic definition of absolute value to find the following values.

Exercises

For the following problems, determine each of the values.

| ( 8 ) |

8

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| ( 1 ) |

1

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| ( 17 | 12 | ) |

5

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| ( 46 | 24 | ) |

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( | 6 | + | 4 | ) 2

100

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( | 4 | + | 6 | ) 2 ( | 2 | ) 3

92

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[ | 10 | 6 ] 2

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[ ( | 4 | + | 3 | ) 3 ] 2

1

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A Mission Control Officer at Cape Canaveral makes the statement "lift-off, T minus 50 seconds." How long before lift-off?

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Due to a slowdown in the industry, a Silicon Valley computer company finds itself in debt $ 2 , 400 , 000 . Use absolute value notation to describe this company's debt.

| $ 2 , 400 , 000 |

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A particular machine is set correctly if upon action its meter reads 0 units. One particular machine has a meter reading of 1.6 upon action. How far is this machine off its correct setting?

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Exercises for review

( [link] ) Write the following phrase using algebraic notation: "four times ( a + b ) ."

4 ( a + b )

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( [link] ) Is there a smallest natural number? If so, what is it?

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( [link] ) Name the property of real numbers that makes 5 + a = a + 5 a true statement.

commutative property of addition

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( [link] ) Find the quotient of x 6 y 8 x 4 y 3 .

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( [link] ) Simplify ( 4 ) .

4

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Questions & Answers

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
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Adin Reply
?
Kyle
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Adin
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Adin
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Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
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research.net
kanaga
sciencedirect big data base
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Introduction about quantum dots in nanotechnology
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nano basically means 10^(-9). nanometer is a unit to measure length.
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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
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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.
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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
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how did you get the value of 2000N.What calculations are needed to arrive at it
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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