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• Slide 1
• Absolute Value 07/27/12lntaylor
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• 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 07/27/12lntaylor
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• LO1: LO2: Define and recognize Absolute Value Simplify and solve Absolute Value problems 07/27/12lntaylor TOC Learning Objectives
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• PK1:Knowledge of number line 07/27/12lntaylor TOC Previous Knowledge PK2:Knowledge of simplification and solution of problems
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• Def1: Def2: Absolute Value is a defined as the distance from 0 Since Absolute Value is a distance, it is always positive 07/27/12lntaylor TOC Absolute Value (Definitions) Def3:Absolute Value is represented by the symbol | | Indicating the value inside the | | is always positive Examples |3| = 3; |-3| = 3; |x| = x; |-x| = x
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• 07/27/12lntaylor TOC Absolute Value Number Line
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• Step1: Step2: Absolute Value is the distance from 0 Find 0 on the number line |2| is always positive (because it is a distance) 07/27/12lntaylor TOC What is the |2|? Step3:Find 2 on the number line Step4:The |2| is 2 012 | + 2 |
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• Step1: Step2: Absolute Value is the distance from 0 Find 0 on the number line |- 2| is always positive (because it is a distance) 07/27/12lntaylor TOC What is the |-2|? Step3:Find - 2 on the number line Step4:The |-2| is 2 0-2 | + 2 |
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• Now you try What is the |-7|? 07/26/12lntaylor TOC
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• Step1: Step2: Absolute Value is the distance from 0 Find 0 on the number line |-7| is always positive (because it is a distance) 07/27/12lntaylor TOC What is the |- 7|? Step3:Find - 7 on the number line Step4:The |- 7| is 7 0- 6-7 | + 7 |
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• 07/27/12lntaylor TOC Simplifying Absolute Value
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• Step1: Step2: Absolute Value is the distance from 0 Do the operation inside the | | symbol first | | is always positive (because it is a distance) 07/27/12lntaylor TOC What is the |9 7|? Step3:Find your answer on the number line Step4:The |9 - 7| is 2 012 | + 2 | 9 7 = 2
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• What is the 3|1-7|? 07/26/12lntaylor TOC
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• Step1: Step2: Absolute Value is the distance from 0 Do the operation inside the | | symbol first 3| | is always positive because it is a + + 07/27/12lntaylor TOC 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
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• Now you try! What is the 4|1- 6|? 07/26/12lntaylor TOC
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• Step1: Step2: Absolute Value is the distance from 0 Do the operation inside the | | symbol first 4| | is always positive because it is a + + 07/27/12lntaylor TOC 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
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• Now you try! What is the |1- 10| - 3? 07/26/12lntaylor TOC
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• Step1: Step2: Absolute Value is the distance from 0 Do the operation inside the | | symbol first | | -3 will be negative because + / - = - 07/27/12lntaylor TOC 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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• 07/27/12lntaylor TOC Practice Simplification
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• 07/27/12 lntaylor 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 - 1 clear answers
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• 07/27/12lntaylor TOC Solving and Graphing Absolute Value Equalities and Inequalities
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• Step1: Step2: Remember there could be a + or a - inside the | | Therefore set up two equations solving for 3 Notice there is no more | | symbol 07/27/12lntaylor TOC 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)
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• Step1: Step2: Remember there could be a + or a - inside the | | Therefore set up two equations solving for 9 Notice there is no more | | symbol but there is ( ) 07/27/12lntaylor TOC 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) = + 93(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
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• Step1: Step2: Remember there could be a + or a - inside the | | Therefore set up two equations solving for the right side Notice there is no more | | symbol but there is ( ) 07/27/12lntaylor TOC 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 means any number less than or equal to 6 and greater than or equal to 14 works 6 x 14
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• Step1:Remember we solved this on the last slide 07/27/12lntaylor TOC Graph the solutions to x in 2|x 10| 8 Step2:Draw a number line that includes your solution Step3:6 x 14 means any number less than or equal to 6 and greater than or equal to 14 works 6 x 14 Step4: and use colored in dots use open dots 6 14 Step5:Draw your solutions
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• Now you try! Graph the solutions to 2|x+3| < 30 07/26/12lntaylor TOC
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• Step1: Step2: Remember there could be a + or a - inside the | | Therefore set up two equations solving for the right side Notice there is no more | | symbol but there is ( ) 07/27/12lntaylor TOC 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 means any 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
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• Now you try! Graph the solutions to |x+3| 5 < 30 07/26/12lntaylor TOC
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• Step1: Step2: Remember there could be a + or a - inside the | | Therefore set up two equations solving for the right side Notice there is no more | | symbol but there is ( ) 07/27/12lntaylor TOC 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 means any 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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• 07/27/12lntaylor TOC Practice Absolute Value Equalities and Inequalities
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• 07/27/12lntaylor TOC ProblemAnswer 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 3x or (6x 3x) - 13 < x < 7 > - 16 - 4 > -1 x 0 > clear answers - 2 0 ##### Section 8.3 Absolute Value Functions. 8.3 Lecture Guide: Absolute Value Functions Objective 1: Sketch the graph of an absolute value function
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