Absolute Value 07/27/12lntaylor ©

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 ©

LO1: LO2: Define and recognize Absolute Value Simplify and solve Absolute Value problems 07/27/12lntaylor © TOC Learning Objectives

PK1:Knowledge of number line 07/27/12lntaylor © TOC 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 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

07/27/12lntaylor © TOC Absolute Value Number Line

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 |

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 |

Now you try What is the |-7|? 07/26/12lntaylor © TOC

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 |

07/27/12lntaylor © TOC Simplifying Absolute Value

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 | 9 – 7 = 2

What is the 3|1-7|? 07/26/12lntaylor © TOC

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 | 3|1 – 7| = 3|- 6| = 3 ∗ 6 = 18

Now you try! What is the 4|1- 6|? 07/26/12lntaylor © TOC

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 | 4|1 – 6| = 4|- 5| = 4 ∗5 = 20

Now you try! What is the |1- 10| ÷ - 3? 07/26/12lntaylor © TOC

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 | |1 – 10| = |- 9| ÷ -3 = +9/-3 = -3

07/27/12lntaylor © TOC Practice Simplification

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 > > > > > > - 3x > - 5 > 60 > > > > > x - 1 clear answers

07/27/12lntaylor © TOC 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 ± 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 = – x = = x = x - 2 = x 1 = x = 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 ± 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) = – 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 = = x 13 = x 3|10 – x| = 9 | | = 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 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 – ≥ x ≥ Step5:6 ≥ x ≥ 14 means any number less than or equal to 6 and greater than or equal to 14 works 6 ≥ x ≥ 14

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

Now you try! Graph the solutions to 2|x+3| < 30 07/26/12lntaylor © TOC

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

Now you try! Graph the solutions to |x+3| ÷ 5 < 30 07/26/12lntaylor © TOC

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 Step6:Yes it works; now graph the solution set

07/27/12lntaylor © TOC Practice Absolute Value Equalities and Inequalities

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 and 51 3x or (6x – 3x) - 13 < x < 7 > > -1 ≤ x ≤ 0 > clear answers - 2 0