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
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LO1:
LO2:
Define and recognize Absolute Value
Simplify and solve Absolute Value problems
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Learning Objectives
PK1: Knowledge of number line
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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
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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
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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)
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What is the |2|?
Step3: Find 2 on the number line
Step4: The |2| is 2
0 1 2
| + 2 |
Step1:
Step2:
Absolute Value is the distance from 0Find 0 on the number line
|- 2| is always positive (because it is a distance)
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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|?
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Step1:
Step2:
Absolute Value is the distance from 0Find 0 on the number line
|-7| is always positive (because it is a distance)
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What is the |- 7|?
Step3: Find - 7 on the number line
Step4: The |- 7| is 7
0- 6-7
| + 7 |
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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)
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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|?
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Step1:
Step2:
Absolute Value is the distance from 0Do the operation inside the | | symbol first
3| | is always positive because it is a + ∗ +
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What is 3|1 – 7|?
Step3: Find your answer on the number line
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|?
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Step1:
Step2:
Absolute Value is the distance from 0Do the operation inside the | | symbol first
4| | is always positive because it is a + ∗ +
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What is 4|1 – 6|?
Step3: Find your answer on the number line
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?
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Step1:
Step2:
Absolute Value is the distance from 0Do the operation inside the | | symbol first
| | ÷ -3 will be negative because + / - = -
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What is |1 – 10| ÷ - 3 ?
Step3: Find your answer on the number line
Step4: The |1 - 10| ÷ - 3 = - 3
0- 1 - 3
| - 3 |
|1 – 10| = |- 9| ÷ -3 = +9/-3 = -3
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Practice Simplification
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| | 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
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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
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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:
Remember there could be a + or a - inside the | |
Therefore set up two equations solving for ± 9Notice there is no more | | symbol but there is ( )
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What is x if 3|10 – x| = 9?
Step3: Solve each equation for x; watch your signs!!!!
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:
Remember there could be a + or a - inside the | |
Therefore set up two equations solving for the right sideNotice there is no more | | symbol but there is ( )
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What is x if 2|x – 10| ≥ 8?
Step3: Solve each equation for x; watch your signs!!!!
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
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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
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Step1:
Step2:
Remember there could be a + or a - inside the | |
Therefore set up two equations solving for the right sideNotice there is no more | | symbol but there is ( )
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Graph the solutions to 2|x+3| < 30
Step3: Solve each equation for x; watch your signs!!!!
– 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
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Step1:
Step2:
Remember there could be a + or a - inside the | |
Therefore set up two equations solving for the right sideNotice there is no more | | symbol but there is ( )
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Graph the solutions to |x+3| ÷ 5 < 30
Step3: Solve each equation for x; watch your signs!!!!
– 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
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Practice Absolute Value Equalities and Inequalities
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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