2 6 Equations and Inequalities Involving Absolute Value

2. 6 Equations and Inequalities Involving Absolute Value By looking at the equation, what number could you substitute for x and the absolute value of that number would give 5? x = 5 or x = – 5 would satisfy the equation. Or we could think in terms of distance. What numbers are 5 units away from zero? The numbers are 5 and – 5. Bobs. Math. Class. Com Copyright © 2010 All Rights Reserved. 1

2. 6 Equations and Inequalities Involving Absolute Value We can use the following property to solve equations with absolute values. Property 1 Using the above property, we have: Please check to verify. Your Turn Problem #1 Bobs. Math. Class. Com Copyright © 2010 All Rights Reserved. 2

2. 6 Equations and Inequalities Involving Absolute Value Solution: To use property 1, the absolute value must be by itself on the left hand side. Therefore, add 3 to both sides before using the property 1. Now use the property given. Your Turn Problem #2 Bobs. Math. Class. Com Copyright © 2010 All Rights Reserved. 3

2. 6 Equations and Inequalities Involving Absolute Value Inequalities with Absolute Values What values for x would satisfy the inequality? The numbers that satisfy this inequality are all the real numbers smaller than 4 and greater than – 4 (i. e. -3, -2, -1, 0, 1, 2, 3, -1. 4, 3. 78 etc). We could then write x < 4 and x > – 4 The graph would be the interval (– 4, 4). ( ) -4 4 We can use the following property to solve inequalities with absolute values. Property 2 Note: The first inequality is written exactly the same without the absolute value bars. In the second inequality, the direction of the inequality has changed and a negative sign is inserted on the right hand side. Also important is the word “and” between the two inequalities. This implies we want the intersection of the two sets. Bobs. Math. Class. Com Copyright © 2010 All Rights Reserved. 4

2. 6 Equations and Inequalities Involving Absolute Value 1. Write as two inequalities (property 3) and solve. 2. Graph the compound inequality. 3. Write in interval notation. ( ) -2 3 Therefore, the solution set is (– 2, 3) Your Turn Problem #3 ( ) -4/3 2 Bobs. Math. Class. Com Copyright © 2010 All Rights Reserved. 5

2. 6 Equations and Inequalities Involving Absolute Value What values for x would satisfy the inequality? The numbers that satisfy this inequality are all the real numbers less than – 4 and greater than 4 (i. e. -5, -6, -7, 5, 6, 7, -4. 3, 5. 78 etc). We could then write x < – 4 or x > 4 The graph would be the intervals (– , – 4) (4, ). ) ( -4 4 We have another property to solve inequalities with absolute values. Property 3 Note: The first inequality is written exactly the same without the absolute value bars. In the second inequality, the direction of the inequality has changed and a negative sign is inserted on the right hand side. Also important is the word “or” between the two inequalities. This implies we want the union of the two sets. The procedure for both absolute inequalities is almost the same. If < , use “and”. If >, use “or”. If you forget, the graphing will remind you. Bobs. Math. Class. Com Copyright © 2010 All Rights Reserved. 6

2. 6 Equations and Inequalities Involving Absolute Value 1. Write as two inequalities (property 3) and solve. 2. Graph the compound inequality. 3. Write in interval notation. Therefore the graph is: ) ( -6 3 And interval notation is (– , – 6) (3, ). Your Turn Problem #4 Answer: ) ( -3 8 Bobs. Math. Class. Com Copyright © 2010 All Rights Reserved. (– , – 3) (8, ) 7

2. 6 Equations and Inequalities Involving Absolute Value Notes: If the absolute value is less than a number, use the word “and”. If the absolute value is greater than a number, use the word “or”. If the absolute value is less than a number, the solution will be an interval between two numbers. ( ) If the absolute value is greater than a number, the solution will be two disjoint intervals. ) ( Next Slide Bobs. Math. Class. Com Copyright © 2010 All Rights Reserved. 8

2. 6 Equations and Inequalities Involving Absolute Value Rewrite the absolute value as two inequalities. Since the symbol is less than, use “and”. We also know the graph will be an interval between two numbers. Therefore the graph is: [ ] 0 2/3 Your Turn Problem #5 Answer: ] -8/5 [ 2 Bobs. Math. Class. Com Copyright © 2010 All Rights Reserved. 9

2. 6 Equations and Inequalities Involving Absolute Value To use any of the properties given in this lesson, the absolute value must be by itself on the left hand side before applying the property. Therefore, subtract 3 on both sides first. This step must be done first. Otherwise you will get an incorrect answer. After the necessary step of isolating the absolute value, we can use the properties given. [ ] 2 14 Therefore the graph is: Your Turn Problem #6 Answer: ] -2 [ 3 Bobs. Math. Class. Com Copyright © 2010 All Rights Reserved. 10

2. 6 Equations and Inequalities Involving Absolute Value Solving equations and inequalities with absolute values by inspection In the previous properties, k was defined to be a positive number. If k is a negative number, we can solve the absolute value equation by inspection. The absolute value of (2 x + 5) will be positive number. There is no way that the absolute value of any expression can equal a negative number. Therefore the solution is , (the empty set). Also consider that the absolute value is a distance and distance can not be a negative number. Answer: Your Turn Problem #7 Bobs. Math. Class. Com Copyright © 2010 All Rights Reserved. 11

2. 6 Equations and Inequalities Involving Absolute Value The absolute value will never be a negative number. Therefore the absolute value can not be less than a negative number. There is just no work that is to be done on this problem. If you try to apply property 2 (which you can’t since k is negative), you will get answers which will not be true. Answer: Your Turn Problem #8 Answer: Bobs. Math. Class. Com Copyright © 2010 All Rights Reserved. 12

2. 6 Equations and Inequalities Involving Absolute Value The absolute value will always be greater than a negative number. Therefore the solution is all real numbers. Your Turn Problem #9 Bobs. Math. Class. Com Copyright © 2010 All Rights Reserved. 13