Absolute Value 07/27/12lntaylor ©. Table of Contents Learning Objectives Absolute Value Number Line Simplifying Absolute Value Practice Simplifying Absolute

Absolute Value

07/27/12 lntaylor ©

Table of Contents

Learning Objectives

Absolute Value Number Line

Simplifying Absolute Value

Practice Simplifying Absolute Values

Solving and Graphing Absolute Value Equalities and Inequalities

Practice Solving Absolute Value Problems

07/27/12 lntaylor ©

LO1:

LO2:

Define and recognize Absolute Value

Simplify and solve Absolute Value problems

07/27/12 lntaylor ©TOC

Learning Objectives

PK1: Knowledge of number line


Previous Knowledge

PK2: Knowledge of simplification and solution of problems

Def1:

Def2:

Absolute Value is a defined as “the distance from 0”

Since Absolute Value is a distance, it is always positive


Absolute Value(Definitions)

Def3: Absolute Value is represented by the symbol | | Indicating the value inside the | | is always positiveExamples |3| = 3; |-3| = 3; |x| = x; |-x| = x


Absolute Value Number Line

Step1:

Step2:

Absolute Value is the distance from 0Find 0 on the number line

|2| is always positive (because it is a distance)


What is the |2|?

Step3: Find 2 on the number line

Step4: The |2| is 2

0 1 2

| + 2 |

Step1:

Step2:


|- 2| is always positive (because it is a distance)


What is the |-2|?

Step3: Find - 2 on the number line

Step4: The |-2| is 2

0-1-2

| + 2 |

Now you try

What is the |-7|?


Step1:

Step2:


|-7| is always positive (because it is a distance)


What is the |- 7|?

Step3: Find - 7 on the number line

Step4: The |- 7| is 7

0- 6-7

| + 7 |


Simplifying Absolute Value

Step1:

Step2:

Absolute Value is the distance from 0Do the operation inside the | | symbol first

| | is always positive (because it is a distance)


What is the |9 – 7|?

Step3: Find your answer on the number line

Step4: The |9 - 7| is 2

0 1 2

| + 2 |

9 – 7 = 2

What is the 3|1-7|?


Step1:

Step2:


3| | is always positive because it is a + ∗ +


What is 3|1 – 7|?


Step4: The 3|1 - 7| is 18

0-17-18

| + 18 |

3|1 – 7| = 3|- 6| = 3∗6 = 18

Now you try!

What is the 4|1- 6|?


Step1:

Step2:


4| | is always positive because it is a + ∗ +


What is 4|1 – 6|?


Step4: The 4|1 - 6| is 20

0-19-20

| + 20 |

4|1 – 6| = 4|- 5| = 4∗5 = 20

Now you try!

What is the |1- 10| ÷ - 3?


Step1:

Step2:


| | ÷ -3 will be negative because + / - = -


What is |1 – 10| ÷ - 3 ?


Step4: The |1 - 10| ÷ - 3 = - 3

0- 1 - 3

| - 3 |

|1 – 10| = |- 9| ÷ -3 = +9/-3 = -3


Practice Simplification

07/27/12lntaylor © TOC

| | Equivalents

|2|

|-2|

|2 – 5|

|5 – 2|

|20-15|

|15-20|

-|3x|

- |2-7|

|30 – 90|

4|3|

-3x|6-9|

|

- 4x|-3x|

|- 4| - 5

> 2

>

>

>

>

>

2

3

3

5

5

> - 3x

> - 5

> 60

>

>

>

>

>

12

- 9x

𝟏𝟔𝐱𝟐

−𝟏𝟐𝐱𝟐

- 1

clear answers


Solving and Graphing Absolute Value

Equalities and Inequalities

Step1:

Step2:

Remember there could be a + or a - inside the | |

Therefore set up two equations solving for ± 3Notice there is no more | | symbol


What is x if |1 – x| = 3 ?

Step3: Solve each equation for x; watch your signs!!!!

Step4: Substitute and check your work

1 – x = + 3 1 – x = - 3

1 = + 3 + x

1 - 3 = x

- 2 = x

1 = - 3 + x

1 + 3 = x

4 = x

|1 – x| = 3|1- - 2| = 3 |3| = 3 3 = 3|1 – x| = 3|1- 4| = 3 |- 3| = 3 3 = 3

Step5: The values of x where |1 – x| = 3 are (- 2 and 4)

Step1:

Step2:


Therefore set up two equations solving for ± 9Notice there is no more | | symbol but there is ( )


What is x if 3|10 – x| = 9?


Step4: Substitute and check your work

3(10 – x) = + 9 3(10 – x) = - 9

10 – x = + 9/3

10 – x = 3

10 – 3 = x

3|10 – x| = 9|10 - 7| = 3 |3| = 3 3 = 3

Step5: The values of x where 3|10 – x| = 9 are (7 and 13)

7 = x

10 – x = - 9/3

10 – x = - 3

10 + 3 = x

13 = x

3|10 – x| = 9|10 - 13| = 3 |- 3| = 3 3 = 3

Step1:

Step2:


Therefore set up two equations solving for the right sideNotice there is no more | | symbol but there is ( )


What is x if 2|x – 10| ≥ 8?


Step4: Remember 6 ≥ x means x ≤ 6

– 8 ≥ 2(x – 10) ≥ + 8

– 8/2 ≥ x – 10 ≥ + 8/2

– 4 ≥ x – 10 ≥ 4

– 4 + 10 ≥ x ≥ 4 + 10

Step5: 6 ≥ x ≥ 14 meansany number less than or equal to 6and greater than or equal to 14 works

6 ≥ x ≥ 14

Step1: Remember we solved this on the last slide


Graph the solutions to x in 2|x – 10| ≥ 8

Step2: Draw a number line that includes your solution

Step3: 6 ≥ x ≥ 14 meansany number less than or equal to 6and greater than or equal to 14 works

6 ≥ x ≥ 14

Step4: ≥ and ≤ use colored in dots< and > use open dots

6 14

Step5: Draw your solutions

Now you try!

Graph the solutions to 2|x+3| < 30


Step1:

Step2:




Graph the solutions to 2|x+3| < 30


– 30 < 2(x + 3) < + 30

– 30/2 < x + 3 < + 30/2

– 15 < x + 3 < 15

– 15 – 3 < x < 15 – 3

Step4: – 18 < x < 12 meansany number between, but not including, – 18 and 12 works

– 18 < x < 12

Step5: 0 is between – 18 and 12; see if it works

2|x+3| < 30

2|0+3| < 30

2|3| < 30

6 < 30

Step6: Yes it works; now graph the solution set

-18 0 12

Now you try!

Graph the solutions to |x+3| ÷ 5 < 30


Step1:

Step2:




Graph the solutions to |x+3| ÷ 5 < 30


– 30 < (x + 3) / 5 < + 30

– 30 5 < x + 3 < + 30 5∗ ∗

– 150 < x + 3 < 150

– 150 – 3 < x < 150 – 3

Step4: – 153 < x < 147 meansany number between, but not including, – 18 and 12 works

– 153 < x < 147

Step5: 97 is between – 153 and 147; see if it works

|x+3|/5 < 30

|97+3|/5 < 30

|100|/5 < 30

20 < 30

Step6: Yes it works; now graph the solution set

-153 0 147


Practice Absolute Value Equalities and Inequalities


Problem Answer

What is the absolute value of -|3|?

What is the solution to 2|x+1| = 6?

What is the solution to |x-1|/5 = 10

Simplify 2x|-3x/-6|?

Simplify 3x|-2| - |3x|

What is the solution to |x+3| < 10?

What is the solution to 2|x+10| > 12?

What is the solution to |2x + 1| ≤ 1?

Graph the solution set to |2x + 2| ≤ 2

> - 3 >

>

>

>

>

-4 and 2- 49 and 51

x2

3x or (6x – 3x)

- 13 < x < 7

> - 16 < x and x > - 4

> -1 ≤ x ≤ 0

>

clear answers

- 2 0

Documents

Absolute Value 07/27/12lntaylor ©. Table of Contents Learning Objectives Absolute Value Number Line Simplifying Absolute Value Practice Simplifying Absolute