OpenStax-CNX

You are here: Home » Content » Fundamentals of Mathematics » Absolute Value

• Preface
• Acknowledgements

Lenses

What is a lens?

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

Endorsed by (What does "Endorsed by" mean?)

This content has been endorsed by the organizations listed. Click each link for a list of all content endorsed by the organization.
• CCQ

This collection is included in aLens by: Community College of Qatar

"Used as supplemental materials for developmental math courses."

Click the "CCQ" link to see all content they endorse.

Click the tag icon to display tags associated with this content.

• College Open Textbooks

This collection is included inLens: Community College Open Textbook Collaborative
By: CC Open Textbook Collaborative

"Reviewer's Comments: 'I would recommend this text for a basic math course for students moving on to elementary algebra. The information in most chapters is useful, very clear, and easily […]"

Click the "College Open Textbooks" link to see all content they endorse.

Click the tag icon to display tags associated with this content.

Affiliated with (What does "Affiliated with" mean?)

This content is either by members of the organizations listed or about topics related to the organizations listed. Click each link to see a list of all content affiliated with the organization.
• Featured Content

This collection is included inLens: Connexions Featured Content
By: Connexions

"Fundamentals of Mathematics is a work text that covers the traditional topics studied in a modern prealgebra course, as well as topics of estimation, elementary analytic geometry, and […]"

Click the "Featured Content" link to see all content affiliated with them.

Click the tag icon to display tags associated with this content.

Also in these lenses

• UniqU content

This collection is included inLens: UniqU's lens
By: UniqU, LLC

Click the "UniqU content" link to see all content selected in this lens.

Recently Viewed

This feature requires Javascript to be enabled.

Tags

(What is a tag?)

These tags come from the endorsement, affiliation, and other lenses that include this content.

Inside Collection (Textbook):

Textbook by: Denny Burzynski, Wade Ellis. E-mail the authors

Absolute Value

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: 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 aa size 12{a} {}, denoted aa 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.

Example 1

4=44=4 size 12{ lline 4 rline =4} {}

Example 2

4=44=4 size 12{ lline - 4 rline =4} {}

Example 3

0=00=0 size 12{ lline 0 rline =0} {}

Example 4

5=55=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.

Example 5

3=33=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 -(3)=-3.

Practice Set A

By reasoning geometrically, determine each absolute value.

Exercise 1

77 size 12{ lline 7 rline } {}

7

Exercise 2

33 size 12{ lline - 3 rline } {}

3

Exercise 3

1212 size 12{ lline "12" rline } {}

12

Exercise 4

00 size 12{ lline 0 rline } {}

0

Exercise 5

99 size 12{ - lline 9 rline } {}

-9

Exercise 6

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

-6

Algebraic Definition of Absolute Value

From the problems in Section 3, 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 |a|= a, if  a 0 -a, if < 0

The algebraic definition takes into account the fact that the number aa size 12{a} {} could be either positive or zero a0a0 size 12{ left (a >= 0 right )} {} or negative a<0a<0 size 12{ left (a<0 right )} {}.

1. If the number aa size 12{a} {} is positive or zero a0a0 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.

Note:

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.

Example 6

88 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 8 = 8 size 12{ lline 8 rline =8} {}

Example 7

33 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 33 size 12{ - left ( - 3 right )} {}. By the definition of absolute value and the double-negative property,

3 = 3 = 3 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.

Exercise 7

77 size 12{ lline 7 rline } {}

7

Exercise 8

99 size 12{ lline 9 rline } {}

9

Exercise 9

1212 size 12{ lline - "12" rline } {}

12

Exercise 10

55 size 12{ lline - 5 rline } {}

5

Exercise 11

88 size 12{ - lline 8 rline } {}

-8

Exercise 12

11 size 12{ - lline 1 rline } {}

-1

Exercise 13

5252 size 12{ - lline - "52" rline } {}

-52

Exercise 14

3131 size 12{ - lline - 31 rline } {}

-31

Exercises

Determine each of the values.

Exercise 15

55 size 12{ lline 5 rline } {}

5

Exercise 16

33 size 12{ lline 3 rline } {}

Exercise 17

66 size 12{ lline 6 rline } {}

6

Exercise 18

99 size 12{ lline -9 rline } {}

Exercise 19

11 size 12{ lline -1 rline } {}

1

Exercise 20

44 size 12{ lline -4 rline } {}

Exercise 21

33 size 12{- lline 3 rline } {}

-3

Exercise 22

77 size 12{- lline 7 rline } {}

Exercise 23

- 14- 14 size 12{- lline –14 rline } {}

-14

Exercise 24

00 size 12{ lline 0 rline } {}

Exercise 25

2626 size 12{ lline -"26" rline } {}

26

Exercise 26

2626 size 12{- lline -"26" rline } {}

Exercise 27

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

4

Exercise 28

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

Exercise 29

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

6

Exercise 30

4242 size 12{- left (- lline -"42" rline right )} {}

Exercise 31

5252 size 12{ lline 5 rline - lline -2 rline } {}

3

Exercise 32

2323 size 12{ lline -2 rline rSup { size 8{3} } } {}

Exercise 33

2323 size 12{ lline - left (2 cdot 3 right ) rline } {}

6

Exercise 34

2929 size 12{ lline -2 rline - lline -9 rline } {}

Exercise 35

6+426+42 size 12{ left ( lline -6 rline + lline 4 rline right ) rSup { size 8{2} } } {}

100

Exercise 36

113113 size 12{ left ( lline -1 rline - lline 1 rline right ) rSup { size 8{3} } } {}

Exercise 37

4+62234+6223 size 12{ left ( lline 4 rline + lline -6 rline right ) rSup { size 8{2} } - left ( lline -2 rline right ) rSup { size 8{3} } } {}

92

Exercise 38

- - 10 - 6 2 - - 10 - 6 2

Exercise 39

{4+33}2{4+33}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

Exercise 40

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

Exercise 42

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

Exercises for Review

Exercise 43

((Reference)) Find the sum: 970+521+815970+521+815 size 12{ { {9} over {"70"} } + { {5} over {"21"} } + { {8} over {"15"} } } {}.

Solution

910910 size 12{ { {9} over {"10"} } } {}

Exercise 44

((Reference)) Find the value of 310+4121920310+4121920 size 12{ { { { {3} over {"10"} } + { {4} over {"12"} } } over { { {"19"} over {"20"} } } } } {}.

Exercise 45

((Reference)) Convert 3.2353.235 size 12{3 "." 2 { {3} over {5} } } {} to a fraction.

Solution

31350 or 1635031350 or 16350 size 12{3 { {"13"} over {"50"} } " or " { {"163"} over {"50"} } } {}

Exercise 46

((Reference)) The ratio of acid to water in a solution is 3838 size 12{ { {3} over {8} } } {}. How many mL of acid are there in a solution that contain 112 mL of water?

Exercise 47

((Reference)) Find the value of 6(8)6(8) size 12{-6- $$-8$$ } {}.

2

Content actions

PDF | EPUB (?)

What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

Collection to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks

Module to:

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

Definition of a lens

Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

Who can create a lens?

Any individual member, a community, or a respected organization.

What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks