Solving Open Sentences Involving Absolute Value

1. Solving Open Sentences Involving Absolute Value | | | | | | | | | | | | | | | | | | | | – 3 – 2 – 1 0 1 2 3 4 5 6 – 5 – 4 – 3 – 2 – 1 0 1 2 3 4

2. Section 6-5 Solving Open sentences involving absolute vale

3. Solving Open Sentences Involving Absolute Value There are three types of open sentences that can involve absolute value. Solving Open Sentences Involving Absolute Value Consider the case | x | = n. | x | = 5 means the distance between 0 and x is 5 units If | x | = 5, then x = – 5 or x = 5. The solution set is {– 5, 5}.

4. Solving Open Sentences Involving Absolute Value When solving equations that involve absolute value, there are two cases to consider: Case 1 The value inside the absolute value symbols is positive. Case 2 The value inside the absolute value symbols is negative. Equations involving absolute value can be solved by graphing them on a number line or by writing them as a compound sentence and solving it.

5. means that the distance between b and –6 is 5 units. To find b on the number line, start at –6 and move 5 units in either direction. Answer: The solution set is Solve an Absolute Value Equation Method 1 Graphing The distance from –6 to –11 is 5 units. The distance from –6 to –1 is 5 units.

6. Write as or Original inequality Case 1 Case 2 Subtract 6 from eachside. Simplify. Answer: The solution set is Solve an Absolute Value Equation Example 5-1a Method 2 Compound Sentence

7. Solve an Absolute Value Equation Answer: {12, –2}

8. So, an equation is . Write an Absolute Value Equation Write an equation involving the absolute value for the graph. Find the point that is the same distance from –4 as the distance from 6. The midpoint between –4 and 6 is 1. The distance from 1 to –4 is 5 units. The distance from 1 to 6 is 5 units.

9. Check Substitute –4 and 6 into Answer: Write an Absolute Value Equation

10. Answer: Write an Absolute Value Equation Write an equation involving the absolute value for the graph.

11. Solving Open Sentences Involving Absolute Value Consider the case | x | < n. | x | < 5 means the distance between 0 and x is LESS than 5 units If | x | < 5, then x > – 5 andx < 5. The solution set is {x| – 5 < x < 5}.

12. Solving Open Sentences Involving Absolute Value When solving equations of the form | x | < n, find the intersection of these two cases. Case 1 The value inside the absolute value symbols is less than the positive value of n. Case 2 The value inside the absolute value symbols is greater than negative value of n.

13. Then graph the solution set. Write as and Case 2 Case 1 Original inequality Add 3 to each side. Simplify. Answer: The solution set is Solve an Absolute Value Inequality (<)

14. Then graph the solution set. Answer: Solve an Absolute Value Inequality (<)

15. Solving Open Sentences Involving Absolute Value Consider the case | x | > n. | x | > 5 means the distance between 0 and x is GREATER than 5 units If | x | > 5, then x < – 5 orx > 5. The solution set is {x| x < – 5 or x > 5}.

16. Solving Open Sentences Involving Absolute Value When solving equations of the form | x | > n, find the union of these two cases. Case 1 The value inside the absolute value symbols is greater than the positive value of n. Case 2 The value inside the absolute value symbols is less than negative value of n.

17. Then graph the solution set. Write as or Case 2 Case 1 Original inequality Add 3 to each side. Simplify. Divide each side by 3. Simplify. Solve an Absolute Value Inequality (>)

18. Answer: The solution set is Solve an Absolute Value Inequality (>)

19. Then graph the solution set. Answer: Solve an Absolute Value Inequality (>)

20. Solving Open Sentences Involving Absolute Value In general, there are three rules to remember when solving equations and inequalities involving absolute value: • If then or(solution set of two numbers) • If then and(intersection of inequalities) • If then or(union of inequalities)

21. Assignment • Study Guide 6-5 (In-Class) • Pages 349-350 #’s 14-19, 24-35, 40, 41. (Homework)