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Textbook by: Denny Brown. E-mail the author

Absolute Value

Summary: This document is a basic introduction to absolute value.

When we want to talk about how “large” a number is without regard as to whether it is positive or negative, we use the absolute value function.

Geometric Definition of Absolute Value

The absolute value of a number a, denoted aa size 12{ lline a rline } {}, is the distance from that number to the origin (zero) on the number line. Absolute value answers the question of “how far,” not “which way.” That distance is always given as a nonnegative number. In short:

• If a number is positive (or zero), the absolute value function does nothing to it: 3=33=3 size 12{ lline 3 rline =3} {}
• If a number is negative, the absolute value function makes it positive: 3=33=3 size 12{ lline -3 rline =3} {}

WARNING: If there is arithmetic to do inside the absolute value function, you must do it before taking the absolute value—the absolute value function acts on the result of whatever is inside it. For example, a common error is

5+2=5+2=75+2=5+2=7 size 12{ lline 5+ left (-2 right ) rline =5+2=7} {} (Wrong!)

The mistake here is in assuming that the absolute value makes everything inside it positive. This is not true. It only makes the result positive. The correct result is

5+2=3=35+2=3=3 size 12{ lline 5+ left (-2 right ) rline = lline 3 rline =3} {} (Correct)

Examples

Determine each value.

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

• The answer is 4:

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

• The answer is 4:

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

• The answer is 0.

5=5= size 12{- lline 5 rline ={}} {}__

• 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 ={}} {}__

• 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.

Geometric Absolute Value Exercises

By reasoning geometrically, determine each absolute value.

Exercise 1

7=7= size 12{ lline 7 rline ={}} {}__

7

Exercise 2

3=3= size 12{ lline -3 rline ={}} {}__

3

Exercise 3

12=12= size 12{ lline "12" rline ={}} {}__

12

Exercise 4

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

0

Exercise 5

9=9= size 12{- lline 9 rline ={}} {}__

−9

Exercise 6

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

−6

Algebraic Definition of Absolute Value

From the preceding exercises, we can suggest the following algebraic definition of absolute value. Note that the definition has two parts.

The absolute value of a number a is

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

1. If the number a is positive or zero (a ≥ 0), the first part of the definition applies. The first 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 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.

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.

Examples

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

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

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

33 size 12{ lline -3 rline } {} = __

• 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 definition of absolute value and the double-negative property, 33 size 12{ lline -3 rline } {} = −(−3) = 3

Algebraic Absolute Value Exercises

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

Module Review Exercises

For the following problems, determine each of the values. <<I figured out these answers; still have to QA>> <<also to verify: have squares & cubes been covered in earlier modules? Subtracting negative numbers? >>

Exercise 15

55 size 12{ lline 5 rline } {} = __

5

Exercise 16

33 size 12{ lline 3 rline } {} = __

3

Exercise 17

66 size 12{ lline 6 rline } {} = __

6

Exercise 18

99 size 12{ lline -9 rline } {} = __

9

Exercise 19

11 size 12{ lline -1 rline } {} = __

1

Exercise 20

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

4

Exercise 21

33 size 12{- lline 3 rline } {} = __

-3

Exercise 22

77 size 12{- lline 7 rline } {} = __

-7

Exercise 23

1414 size 12{- lline -"14" rline } {} = __

-14

Exercise 24

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

0

Exercise 25

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

26

Exercise 26

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

-26

Exercise 27

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

4

Exercise 28

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

2

Exercise 29

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

6

Exercise 30

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

42

Exercise 31

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

3

Exercise 32

2222 size 12{ lline -2 rline rSup { size 8{2} } } {} = __

4

Exercise 33

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

-6

Exercise 34

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

-7

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} } } {} = __

0

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

10621062 size 12{- left [ lline -"10" rline -6 right ] rSup { size 8{2} } } {} = __

Solution

-16 (Note that this is different from the answer for 10621062 size 12{ left [- left ( lline -"10" rline -6 right ) right ] rSup { size 8{2} } } {}; pay attention to the order of operations.)

Exercise 39

4+3324+332 size 12{- left lbrace - left [- lline -4 rline + lline -3 rline right ] rSup { size 8{3} } right rbrace rSup { size 8{2} } } {} = __

-1

Exercise 40

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

Solution

5050 size 12{ lline -"50" rline } {} = 50 seconds

Exercise 42

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

Solution

1.61.6 size 12{ lline -1 "." 6 rline } {} = 1.6

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