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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

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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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