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Warm-Up : Describe the similarities and differences between equations and inequalities.

Name: Date: Period: Topic: Solving Absolute Value Equations & Inequalities Essential Question : What is the process needed to solve absolute value equations and inequalities?. Warm-Up : Describe the similarities and differences between equations and inequalities. Home-Learning #2 Review.

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Warm-Up : Describe the similarities and differences between equations and inequalities.

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  1. Name:Date:Period: Topic: Solving Absolute Value Equations & InequalitiesEssential Question: What is the process needed to solve absolute value equations and inequalities? Warm-Up: Describe the similarities and differences between equations and inequalities.

  2. Home-Learning #2 Review

  3. Quiz #7:

  4. Recall : Absolute value |x |: is the distance between x and 0. If |x | = 8, then – 8 and 8 is a solution of the equation ; or |x |  8, then any number between 8 and 8 is a solution of the inequality.

  5. Absolute Value (of x) • Symbol lxl • The distance x is from 0 on the number line. • Always positive • Ex: l-3l=3 -4 -3 -2 -1 0 1 2 Recall: You can solve some absolute-value equations using mental math. For instance, you learned that the equation |x| 3 has two solutions: 3 and 3. To solve absolute-value equations, you can use the fact that the expression inside the absolute value symbols can be either positive or negative.

  6. Solving an Absolute-Value Equation: Solve | x  2 |  5 Solve | 2x  7 |  5  4

  7. Answer :: The expressionx  2 can be equal to 5 or 5. x  2IS NEGATIVE | x  2 |  5 x  2IS POSITIVE x  2  5 | x  2 |  5 x  3 x  2  5 CHECK x  7 Solving an Absolute-Value Equation Solve | x  2 |  5 The equation has two solutions: 7 and –3. | 7  2 |  | 5 |  5 |3  2 |  | 5 |  5

  8. Answer :: Isolate theabsolute value expressionon one side of the equation. 2x  7 IS NEGATIVE 2x  7 IS POSITIVE 2x  7 IS NEGATIVE 2x  7 IS POSITIVE | 2x  7 |  5  4 | 2x  7 |  5  4 | 2x  7 |  9 | 2x  7 |  9 2x  7  9 2x  7  +9 2x  7  9 2x  7  +9 2x  2 x  8 2x  16 x  1 TWO SOLUTIONS Solve | 2x  7 |  5  4 SOLUTION Isolate theabsolute value expressionon one side of the equation. 2x  7 IS POSITIVE 2x  7 IS NEGATIVE | 2x  7 |  5  4 | 2x  7 |  5  4 | 2x  7 |  9 | 2x  7 |  9 2x  7  +9 2x  7  9 2x  16 2x  2 x  1

  9. Solve the following Absolute-Value Equation: Practice: 1) Solve 6x-3 = 15 2) Solve 2x + 7 -3 = 8

  10. Answer :: 1) Solve 6x-3 = 15 6x-3 = 15 or 6x-3 = -15 6x = 18 or 6x = -12 x = 3 or x = -2 * Plug in answers to check your solutions!

  11. Answer :: 2) Solve 2x + 7 -3 = 8 Get the abs. value part by itself first! 2x+7 = 11 Now split into 2 parts. 2x+7 = 11 or 2x+7 = -11 2x = 4 or 2x = -18 x = 2 or x = -9 Check the solutions.

  12. ***Important NOTE*** 3 2x + 9 +12 = 10 - 12 - 12 3 2x + 9 = - 2 3 3 No Solution 2x + 9 = - 2 3 What about this absolute value equation? 3x – 6 – 5 = – 7

  13. Solving & Graphing Absolute Value Inequalities

  14. Solving an Absolute Value Inequality: • Step 1: Rewrite the inequality as a conjunction or a disjunction. • If you have a you are working with a conjunction or an ‘and’ statement. Remember: “Less thand” • If you have ayou are working with a disjunction oran ‘or’ statement. Remember: “Greator” • Step 2: In the second equation you must negate the right hand side and reversethe direction of the inequality sign. • Solve as a compound inequality.

  15. Ex: “and” inequality • Becomes an “and” problem Positive Negative 4x – 9 ≤ 21 4x – 9 ≥ -21 + 9 + 9 + 9 + 9 4x ≤ 30 4x ≥ -12 4 4 4 4 x ≤ 7.5 x ≥ -3 -3 7 8

  16. This is an ‘or’ statement. (Greator). Ex: “or” inequality In the 2nd inequality, reverse the inequality sign and negate the right side value. -4 3 |2x + 1| > 7 2x + 1 > 7 or 2x + 1 < - 7 – 1 - 1 – 1 - 1 2x > 6 2x < - 8 2 2 2 2 x < - 4 x > 3

  17. Solving Absolute Value Inequalities: Solve | x  4 | < 3 and graph the solution. Solve | 2x  1 | 3  6 and graph the solution.

  18. Answer ::  Solve | x  4 | < 3 x  4 IS POSITIVE x  4 IS NEGATIVE | x  4|  3 | x  4|  3 Reverse inequality symbol. x  4  3 x  4  3 x  7 x  1 The solution is all real numbers greater than 1 and less than 7. This can be written as 1 x  7.

  19. Answer :: 2x + 1 IS POSITIVE 2x + 1 IS NEGATIVE | 2x  1 | 3  6 | 2x  1 | 3  6 | 2x  1 |  9 | 2x  1 |  9 2x  1  9 2x  1  +9 2x 10 2x  8 x  4 x 5  6  5  4  3  2  1 0 1 2 3 4 5 6 Solve | 2x  1| 3  6 and graph the solution. Reverse inequality symbol. The solution is all real numbers greater than or equal to 4orless than or equal to 5. This can be written as the compound inequality x  5orx 4.

  20. Solve and graph the following Absolute-Value Inequalities: 3) • |x -5| < 3

  21. Answer :: Solve & graph. 3) • Get absolute value by itself first. • Becomes an “or” problem -2 3 4

  22. Answer :: 2 8 This is an ‘and’ statement. (Less thand). Rewrite. In the 2nd inequality, reverse the inequality sign and negate the right side value. Solve each inequality. Graph the solution. • |x -5|< 3 x -5< 3 and x -5< 3 x -5< 3 and x -5> -3 x < 8 and x > 2 2 < x < 8

  23. Solve and Graph 5) 4m - 5 > 7 or 4m - 5 < - 9 6) 3 < x - 2 < 7 7) |y – 3| > 1 • |p + 2| + 4 < 10 • |3t - 2| + 6 = 2

  24. Home-Learning #3: • Page 211 - 212 (18, 26,36, 40, 64)

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