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Absolute Value

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

Summary: 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 a , denoted | a | | a | , is the distance from a 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:

Example 1

| 4 |=4 | 4 |=4
A number line with arrows on each end, labeled from zero to six in increments of one. There is a horizontal curly brace starting from zero, and ending at four. It is labeled as 'four units in length'.

Example 2

| −4 |=4 | −4 |=4
A number line with arrows on each end, labeled from negative six to zero in increments of one. There is a horizontal curly brace starting from negative four, and ending at zero. It is labeled as 'four units in length'.

Example 3

| 0 |=0 | 0 |=0

Example 4

−| 5 |=−5 −| 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 −5 .

Example 5

−| −3 |=−3 −| −3 |=−3 .
The quantity on the left side of the equal sign is read as "negative the absolute value of −3 −3 ." The absolute value of −3 −3 is 3. Hence, negative the absolute value of −3 −3 is −( 3 )=−3 −( 3 )=−3 .

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 a is

 | a |={ a if a≥0 −a if a<0 | a |={ a if a≥0 −a if a<0

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

  1. If the number a a is positive or zero ( ≥0 ) ( ≥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 a is negative ( <0 ) ( <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.

Example 6

| 8 | | 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 | 8 |=8

Example 7

| −3 | | −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 −3 is the opposite of −3 −3 , which is −( −3 ) −( −3 ) . By the double-negative property, −( −3 )=3 −( −3 )=3 .

| −3 |=3 | −3 |=3

Practice Set A

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

Exercise 1

| 7 | | 7 |

Solution

7

Exercise 2

| 9 | | 9 |

Solution

9

Exercise 3

| −12 | | −12 |

Solution

12

Exercise 4

| −5 | | −5 |

Solution

5

Exercise 5

−| 8 | −| 8 |

Solution

−8 −8

Exercise 6

−| 1 | −| 1 |

Solution

−1 −1

Exercise 7

−| −52 | −| −52 |

Solution

−52 −52

Exercise 8

−| −31 | −| −31 |

Solution

−31 −31

Exercises

For the following problems, determine each of the values.

Exercise 9

| 5 | | 5 |

Solution

5

Exercise 10

| 3 | | 3 |

Exercise 11

| 6 | | 6 |

Solution

6

Exercise 12

| 14 | | 14 |

Exercise 13

| −8 | | −8 |

Solution

8

Exercise 14

| −10 | | −10 |

Exercise 15

| −16 | | −16 |

Solution

16

Exercise 16

−| 8 | −| 8 |

Exercise 17

−| 12 | −| 12 |

Solution

−12 −12

Exercise 18

−| 47 | −| 47 |

Exercise 19

−| 9 | −| 9 |

Solution

−9 −9

Exercise 20

| −9 | | −9 |

Exercise 21

| −1 | | −1 |

Solution

1

Exercise 22

| −4 | | −4 |

Exercise 23

−| 3 | −| 3 |

Solution

−3 −3

Exercise 24

−| 7 | −| 7 |

Exercise 25

−| −14 | −| −14 |

Solution

−14 −14

Exercise 26

−| −19 | −| −19 |

Exercise 27

−| −28 | −| −28 |

Solution

−28 −28

Exercise 28

−| −31 | −| −31 |

Exercise 29

−| −68 | −| −68 |

Solution

−68 −68

Exercise 30

| 0 | | 0 |

Exercise 31

| −26 | | −26 |

Solution

26

Exercise 32

−| −26 | −| −26 |

Exercise 33

−| −(−8) | −| −(−8) |

Solution

−8 −8

Exercise 34

−| −(−4) | −| −(−4) |

Exercise 35

−| −(−1) | −| −(−1) |

Solution

−1 −1

Exercise 36

−| −(−7) | −| −(−7) |

Exercise 37

−( −| 4 | ) −( −| 4 | )

Solution

4

Exercise 38

−( −| 2 | ) −( −| 2 | )

Exercise 39

−( −| −6 | ) −( −| −6 | )

Solution

6

Exercise 40

−( −| −42 | ) −( −| −42 | )

Exercise 41

| −| −3 | | | −| −3 | |

Solution

3

Exercise 42

| −| −15 | | | −| −15 | |

Exercise 43

| −| −12 | | | −| −12 | |

Solution

12

Exercise 44

| −| −29 | | | −| −29 | |

Exercise 45

| 6−| −2 | | | 6−| −2 | |

Solution

4

Exercise 46

| 18−| −11 | | | 18−| −11 | |

Exercise 47

| 5−| −1 | | | 5−| −1 | |

Solution

4

Exercise 48

| 10−| −3 | | | 10−| −3 | |

Exercise 49

| −( 17−| −12 | ) | | −( 17−| −12 | ) |

Solution

5

Exercise 50

| −( 46−| −24 | ) | | −( 46−| −24 | ) |

Exercise 51

| 5 |−| −2 | | 5 |−| −2 |

Solution

3

Exercise 52

| −2 | 3 | −2 | 3

Exercise 53

| −(2⋅3) | | −(2⋅3) |

Solution

6

Exercise 54

| −2 |+| −9 | | −2 |+| −9 |

Exercise 55

( | −6 |+| 4 | ) 2 ( | −6 |+| 4 | ) 2

Solution

100

Exercise 56

( | −1 |-| 1 | ) 3 ( | −1 |-| 1 | ) 3

Exercise 57

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

Solution

92

Exercise 58

− [ | −10 |−6 ] 2 − [ | −10 |−6 ] 2

Exercise 59

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

Solution

−1 −1

Exercise 60

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

Exercise 61

Due to a slowdown in the industry, a Silicon Valley computer company finds itself in debt $2,400,000 $2,400,000 . Use absolute value notation to describe this company's debt.

Solution

| $−2,400,000 | | $−2,400,000 |

Exercise 62

A particular machine is set correctly if upon action its meter reads 0 units. 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 63

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

Solution

4( a+b ) 4( a+b )

Exercise 64

((Reference)) Is there a smallest natural number? If so, what is it?

Exercise 65

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

Solution

commutative property of addition

Exercise 66

((Reference)) Find the quotient of x 6 y 8 x 4 y 3 x 6 y 8 x 4 y 3 .

Exercise 67

((Reference)) Simplify −(−4) −(−4) .

Solution

4

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