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Math Pacing

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  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 Math Pacing

  2. 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}.

  3. Solving Open Sentences Involving Absolute Value When solving equations that involve absolute value, there are two cases to consider: Solving Open Sentences Involving Absolute Value 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.

  4. 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 Example 5-1a Method 1 Graphing The distance from –6 to –11 is 5 units. The distance from –6 to –1 is 5 units.

  5. 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

  6. Solve an Absolute Value Equation Example 5-1b Answer: {12, –2}

  7. So, an equation is . Write an Absolute Value Equation Example 5-2a 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.

  8. Check Substitute –4 and 6 into Answer: Write an Absolute Value Equation Example 5-2a

  9. Answer: Write an Absolute Value Equation Example 5-2b Write an equation involving the absolute value for the graph.

  10. 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 Solving Open Sentences Involving Absolute Value If | x | < 5, then x > – 5 andx < 5. The solution set is {x| – 5 < x < 5}.

  11. Solving Open Sentences Involving Absolute Value When solving equations of the form | x | < n, find the intersection of these two cases. Solving Open Sentences Involving Absolute Value 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.

  12. 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 (<) Example 5-3a

  13. Then graph the solution set. Answer: Solve an Absolute Value Inequality (<) Example 5-3b

  14. 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 Solving Open Sentences Involving Absolute Value If | x | > 5, then x < – 5 orx > 5. The solution set is {x| x < – 5 or x > 5}.

  15. Solving Open Sentences Involving Absolute Value When solving equations of the form | x | > n, find the union of these two cases. Solving Open Sentences Involving Absolute Value 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.

  16. 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 (>) Example 5-4a

  17. Answer: The solution set is Solve an Absolute Value Inequality (>) Example 5-4a

  18. Then graph the solution set. Answer: Solve an Absolute Value Inequality (>) Example 5-4b

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