<< Chapter < Page Chapter >> Page >
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 .

Got questions? Get instant answers now!

| 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 .

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

| 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

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

| ( 1 ) |

1

Got questions? Get instant answers now!

| ( 17 | 12 | ) |

5

Got questions? Get instant answers now!

| ( 46 | 24 | ) |

Got questions? Get instant answers now!

( | 6 | + | 4 | ) 2

100

Got questions? Get instant answers now!

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

92

Got questions? Get instant answers now!

[ | 10 | 6 ] 2

Got questions? Get instant answers now!

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

1

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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 |

Got questions? Get instant answers now!

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?

Got questions? Get instant answers now!

Exercises for review

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

4 ( a + b )

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

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

commutative property of addition

Got questions? Get instant answers now!

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

Got questions? Get instant answers now!

( [link] ) Simplify ( 4 ) .

4

Got questions? Get instant answers now!

Questions & Answers

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
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
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
anyone know any internet site where one can find nanotechnology papers?
Damian Reply
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
Praveena Reply
hi
Loga
what does nano mean?
Anassong Reply
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.
QuizOver Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Elementary algebra' conversation and receive update notifications?

Ask