# Solving Absolute Value Equations and Inequalities.ppt

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### Solving Absolute Value Equations and Inequalities.ppt

• 1. 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?
• 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 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. Recall:
• 6. Solving an Absolute-Value Equation: Solve | x  2 |  5 Solve | 2x  7 |  5  4
• 7. Solving an Absolute-Value Equation | 7  2 |  | 5 |  5 | 3  2 |  | 5 |  5 Solve | x  2 |  5 The expression x  2 can be equal to 5 or 5. x  2 IS POSITIVE | x  2 |  5 x  2  5 x  7 x  3 x  2 IS NEGATIVE | x  2 |  5 x  2  5 The equation has two solutions: 7 and –3. CHECK Answer ::
• 8. Solve | 2x  7 |  5  4 2x  7 IS POSITIVE | 2x  7 |  5  4 | 2x  7 |  9 2x  7  +9 2x  16 2x  7 IS NEGATIVE | 2x  7 |  5  4 | 2x  7 |  9 2x  7  9 2x  2 x  1 Isolate the absolute value expression on one side of the equation. Isolate the absolute value expression on one side of the equation. SOLUTION 2x  7 IS POSITIVE 2x  7  +9 2x  7 IS NEGATIVE 2x  7  9 2x  7 IS POSITIVE | 2x  7 |  5  4 | 2x  7 |  9 2x  7  +9 2x  16 2x  7 IS NEGATIVE | 2x  7 |  5  4 | 2x  7 |  9 2x  7  9 2x  2 TWO SOLUTIONS x  8 x  1 Answer ::
• 9. Solve the following Absolute-Value Equation: Practice: 1) Solve 6x-3 = 15 2) Solve 2x + 7 -3 = 8
• 10. 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! Answer ::
• 11. 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. Answer ::
• 12. ***Important NOTE*** 3 2x + 9 +12 = 10 - 12 - 12 3 2x + 9 = - 2 3 3 2x + 9 = - 2 3 What about this absolute value equation? 3x – 6 – 5 = – 7
• 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 a you are working with a disjunction or an ‘or’ statement. Remember: “Greator” ● Step 2: In the second equation you must negate the right hand side and reverse the direction of the inequality sign. ● Solve as a compound inequality. or   or  
• 15. Ex: “and” inequality • Becomes an “and” problem 21 9 4   x -3 7 8 4x – 9 ≤ 21 + 9 + 9 4x ≤ 30 x ≤ 7.5 4 4 4x – 9 ≥ -21 + 9 + 9 4x ≥ -12 x ≥ -3 4 4 Positive Negative
• 16. |2x + 1| > 7 This is an ‘or’ statement. (Greator). 3 -4 Ex: “or” inequality 2x + 1 > 7 or 2x + 1 < - 7 – 1 - 1 – 1 - 1 2x > 6 2x < - 8 2 2 2 2 x > 3 In the 2nd inequality, reverse the inequality sign and negate the right side value.
• 17. Solve | x  4 | < 3 and graph the solution. Solving Absolute Value Inequalities: Solve | 2x  1 | 3  6 and graph the solution.
• 18. Solve | x  4 | < 3 x  4 IS POSITIVE x  4 IS NEGATIVE | x  4 |  3 x  4  3 x  7 | x  4 |  3 x  4  3 x  1 Reverse inequality symbol.  This can be written as 1  x  7. The solution is all real numbers greater than 1 and less than 7. Answer ::
• 19. Solve | 2x  1 | 3  6 and graph the solution. | 2x  1 |  3  6 | 2x  1 |  9 2x  1  +9 x  4 2x  8 | 2x  1 | 3  6 | 2x  1 |  9 2x  1  9 2x  10 x  5 2x + 1 IS POSITIVE 2x + 1 IS NEGATIVE  6  5  4  3  2  1 0 1 2 3 4 5 6 The solution is all real numbers greater than or equal to 4 or less than or equal to  5. This can be written as the compound inequality x   5 or x  4. Reverse inequality symbol. Answer ::
• 20. 11 3 2 3    x 3) Solve and graph the following Absolute-Value Inequalities: 4) |x -5| < 3
• 21. Solve & graph. • Get absolute value by itself first. • Becomes an “or” problem 11 3 2 3    x 8 2 3   x 8 2 3 or 8 2 3      x x 6 3 or 10 3    x x 2 or 3 10    x x -2 3 4 Answer :: 3)
• 22. 4) |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 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. 8 2 Answer ::
• 23. Solve and Graph 5) 4m - 5 > 7 or 4m - 5 < - 9 6) 3 < x - 2 < 7 7) |y – 3| > 1 8) |p + 2| + 4 < 10 9) |3t - 2| + 6 = 2
• 24. Home-Learning #3: • Page 211 - 212 (18, 26,36, 40, 64)
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