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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses absolute value. By the end of the module students should understand the geometric and algebraic definitions of absolute value.

Section overview

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

Geometric definition of absolute value

Absolute value-geometric approach

Geometric definition of absolute value:
The absolute value of a number a size 12{a} {} , denoted a size 12{ \lline a \rline } {} , is the distance from a to 0 on the number line.

Absolute value answers the question of "how far," and not "which way." The phrase "how far" implies "length" and length is always a nonnegative quantity . Thus, the absolute value of a number is a nonnegative number.

Sample set a

Determine each value.

4 = 4 size 12{ lline 4 rline =4} {}

A number line with hash marks from 0 to 6, with zero to 4 marked as 4 units in length.

4 = 4 size 12{ lline - 4 rline =4} {}

A number line with hash marks from -6 to 0, with -4 to 0 marked as 4 units in length.

0 = 0 size 12{ lline 0 rline =0} {}

5 = 5 size 12{ - lline 5 rline = - 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.

3 = 3 size 12{ - lline - 3 rline = - 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 .

Practice set a

By reasoning geometrically, determine each absolute value.

7 size 12{ lline 7 rline } {}

7

3 size 12{ lline - 3 rline } {}

3

12 size 12{ lline "12" rline } {}

12

0 size 12{ lline 0 rline } {}

0

9 size 12{ - lline 9 rline } {}

-9

6 size 12{ - lline - 6 rline } {}

-6

Algebraic definition of absolute value

From the problems in [link] , we can suggest the following algebraic defini­tion of absolute value. Note that the definition has two parts.

Absolute value—algebraic approach

Algebraic definition of absolute value
The absolute value of a number a is
| a | = a , if  a 0 - a , if < 0

The algebraic definition takes into account the fact that the number a size 12{a} {} could be either positive or zero a 0 size 12{ left (a>= 0 right )} {} or negative a < 0 size 12{ left (a<0 right )} {} .

  1. If the number a size 12{a} {} is positive or zero a 0 size 12{ left (a>= 0 right )} {} , the upper part of the definition applies. The upper part of the definition tells us that if the number enclosed in the absolute value bars is a nonnegative number, the absolute value of the number is the number itself.
  2. The lower 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.
The definition says that the vertical absolute value lines may be elimi­nated only if we know whether the number inside is positive or negative.

Sample set b

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

8 size 12{ lline 8 rline } {} . The number enclosed within the absolute value bars is a nonnegative number, so the upper part of the definition applies. This part says that the absolute value of 8 is 8 itself.

8 = 8 size 12{ lline 8 rline =8} {}

3 size 12{ lline - 3 rline } {} . The number enclosed within absolute value bars is a negative number, so the lower part of the definition applies. This part says that the absolute value of -3 is the opposite of -3, which is 3 size 12{ - left ( - 3 right )} {} . By the definition of absolute value and the double-negative property,

3 = 3 = 3 size 12{ lline - 3 rline = - left ( - 3 right )=3} {}

Practice set b

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

7 size 12{ lline 7 rline } {}

7

9 size 12{ lline 9 rline } {}

9

12 size 12{ lline - "12" rline } {}

12

5 size 12{ lline - 5 rline } {}

5

8 size 12{ - lline 8 rline } {}

-8

1 size 12{ - lline 1 rline } {}

-1

52 size 12{ - lline - "52" rline } {}

-52

31 size 12{ - lline - 31 rline } {}

-31

Exercises

Determine each of the values.

5 size 12{ lline 5 rline } {}

5

3 size 12{ lline 3 rline } {}

6 size 12{ lline 6 rline } {}

6

9 size 12{ lline -9 rline } {}

1 size 12{ lline -1 rline } {}

1

4 size 12{ lline -4 rline } {}

3 size 12{- lline 3 rline } {}

-3

7 size 12{- lline 7 rline } {}

- 14 size 12{- lline –14 rline } {}

-14

0 size 12{ lline 0 rline } {}

26 size 12{ lline -"26" rline } {}

26

26 size 12{- lline -"26" rline } {}

4 size 12{- left (- lline 4 rline right )} {}

4

2 size 12{- left (- lline 2 rline right )} {}

6 size 12{- left (- lline -6 rline right )} {}

6

42 size 12{- left (- lline -"42" rline right )} {}

5 2 size 12{ lline 5 rline - lline -2 rline } {}

3

2 3 size 12{ lline -2 rline rSup { size 8{3} } } {}

2 3 size 12{ lline - left (2 cdot 3 right ) rline } {}

6

2 9 size 12{ lline -2 rline - lline -9 rline } {}

6 + 4 2 size 12{ left ( lline -6 rline + lline 4 rline right ) rSup { size 8{2} } } {}

100

1 1 3 size 12{ left ( lline -1 rline - lline 1 rline right ) rSup { size 8{3} } } {}

4 + 6 2 2 3 size 12{ left ( lline 4 rline + lline -6 rline right ) rSup { size 8{2} } - left ( lline -2 rline right ) rSup { size 8{3} } } {}

92

- - 10 - 6 2

{ 4 + 3 3 } 2 size 12{- left lbrace left none - left [- lline -4 rline + lline -3 rline right ] rSup { size 8{3} } right rbrace right none rSup { size 8{2} } } {}

-1

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

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 size 12{-$ lline -2,"400","000" rline } {}

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

Exercises for review

( [link] ) Find the sum: 9 70 + 5 21 + 8 15 size 12{ { {9} over {"70"} } + { {5} over {"21"} } + { {8} over {"15"} } } {} .

9 10 size 12{ { {9} over {"10"} } } {}

( [link] ) Find the value of 3 10 + 4 12 19 20 size 12{ { { { {3} over {"10"} } + { {4} over {"12"} } } over { { {"19"} over {"20"} } } } } {} .

( [link] ) Convert 3 . 2 3 5 size 12{3 "." 2 { {3} over {5} } } {} to a fraction.

3 13 50 or 163 50 size 12{3 { {"13"} over {"50"} } " or " { {"163"} over {"50"} } } {}

( [link] ) The ratio of acid to water in a solution is 3 8 size 12{ { {3} over {8} } } {} . How many mL of acid are there in a solution that contain 112 mL of water?

( [link] ) Find the value of 6 ( 8 ) size 12{-6- \( -8 \) } {} .

2

Questions & Answers

can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
Nancy Reply
rolling four fair dice and getting an even number an all four dice
ramon Reply
Kristine 2*2*2=8
Bridget Reply
Differences Between Laspeyres and Paasche Indices
Emedobi Reply
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
Mary Reply
is it 3×y ?
Joan Reply
J, combine like terms 7x-4y
Bridget Reply
im not good at math so would this help me
Rachael Reply
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
how do you translate this in Algebraic Expressions
linda Reply
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris 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
what is nanomaterials​ and their applications of sensors.
Ramkumar Reply
what is nano technology
Sravani Reply
what is system testing?
AMJAD
preparation of nanomaterial
Victor Reply
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
Himanshu Reply
good afternoon madam
AMJAD
what is system testing
AMJAD
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
Prasenjit Reply
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
Ali Reply
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
bamidele 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, Precalculus with engineering applications. OpenStax CNX. Jan 24, 2011 Download for free at http://cnx.org/content/col11267/1.3
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