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Chapter 1: Tools of Algebra 1-5: Absolute Value Equations and Inequalities

Chapter 1: Tools of Algebra 1-5: Absolute Value Equations and Inequalities. Essential Question: What is the procedure used to solve an absolute value equation of inequality? (Tomorrow). 1-5: Absolute Value Equations and Inequalities.

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Chapter 1: Tools of Algebra 1-5: Absolute Value Equations and Inequalities

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  1. Chapter 1: Tools of Algebra1-5: Absolute Value Equations and Inequalities Essential Question: What is the procedure used to solve an absolute value equation of inequality? (Tomorrow)

  2. 1-5: Absolute Value Equations and Inequalities • Absolute Value Equations have two solutions, because the quantity inside the absolute value sign can be positive or negative • Like compound inequalities, create two equations, and solve them independently. • Get the absolute value portion alone • Set the absolute value portion equal to both the positive and negative

  3. 1-5: Absolute Value Equations and Inequalities • Example: Solve |2y – 4| = 12

  4. 1-5: Absolute Value Equations and Inequalities • Example: Solve |2y – 4| = 12

  5. 1-5: Absolute Value Equations and Inequalities • Example: Solve |2y – 4| = 12 • y = 8 or y = -4 • Check: • |2(8) – 4| = |16 – 4| = |12| = 12 • |2(-4) – 4| = |-8 – 4| = |-12| = 12

  6. 1-5: Absolute Value Equations and Inequalities • Multiple Step Absolute Value Equations • Example 2: Solve 3|4w – 1| – 5 = 10 • Get the absolute value portion alone • 3|4w – 1| – 5 = 10

  7. 1-5: Absolute Value Equations and Inequalities • Multiple Step Absolute Value Equations • Example 2: Solve 3|4w – 1| – 5 = 10 • Get the absolute value portion alone • 3|4w – 1| – 5 = 10 + 5 +5 • 3|4w – 1| = 15

  8. 1-5: Absolute Value Equations and Inequalities • Multiple Step Absolute Value Equations • Example 2: Solve 3|4w – 1| – 5 = 10 • Get the absolute value portion alone • 3|4w – 1| – 5 = 10 + 5 +5 • 3|4w – 1| = 153 3 • |4w – 1| = 5 • Now we can split into two equations, just like the last problem

  9. 1-5: Absolute Value Equations and Inequalities • |4w – 1| = 5

  10. 1-5: Absolute Value Equations and Inequalities • |4w – 1| = 5

  11. 1-5: Absolute Value Equations and Inequalities • |4w – 1| = 5 • w = 1.5 or w = -1 • Check (use the original problem): • 3|4(1.5) – 1| – 5 = 3|6 – 1| – 5 = 3|5| – 5 = 3(5) – 5 = 15 – 5 = 10 • 3|4(-1) – 1| – 5 = 3|-4 – 1| – 5 = 3|-5| – 5 = 3(5) – 5 = 15 – 5 = 10

  12. 1-5: Absolute Value Equations and Inequalities • Checking for Extraneous Solutions • Sometimes, we’ll get a solution algebraically that fails when we try and check it. These solutions are called extraneous solutions. • Example 3: Solve |2x + 5| = 3x + 4 • Is the absolution value portion alone? Yes • When we split this into two equations, we have to negate the entire right side of the equation

  13. 1-5: Absolute Value Equations and Inequalities • |2x + 5| = 3x + 4

  14. 1-5: Absolute Value Equations and Inequalities • |2x + 5| = 3x + 4

  15. 1-5: Absolute Value Equations and Inequalities • |2x + 5| = 3x + 4

  16. 1-5: Absolute Value Equations and Inequalities • |2x + 5| = 3x + 4 • x = 1 or x = -1.8 • You’ll have to check your solutions (next slide)

  17. 1-5: Absolute Value Equations and Inequalities • |2x + 5| = 3x + 4 • x = 1 • |2(1) + 5| = 3(1) + 4 • |2 + 5| = 3 + 4 • |7| = 7 (good) • x = -1.8 • |2(-1.8) + 5| = 3(-1.8) + 4 • |-3.6 + 5| = -5.4 + 4 • |1.4| = -1.4 (bad) • The only solution is x = 1 • -1.8 is an extraneous solution.

  18. 1-5: Absolute Value Equations and Inequalities • Assignment • Page 36 • Problems 1 – 15 (all) • You will have to check your solutions for problems 10-15, so show work and identify any extraneous solutions

  19. Chapter 1: Tools of Algebra1-5: Absolute Value Equations and InequalitiesDay 2 Essential Question: What is the procedure used to solve an absolute value equation of inequality?

  20. 1-5: Absolute Value Equations and Inequalities • When we solved absolute value equations, we got the absolute value section alone, and set two equations • One as normal • One where we flipped everything outside the absolute value • When solving absolute value inequalities, we do the same thing, except in addition to flipping everything on the other side of the absolute value, flip the inequality as well • The two lines will always either split apart (greater than) or come together (less than)

  21. 1-5: Absolute Value Equations and Inequalities • Example: Solve |3x + 6| > 12. Graph the solution.

  22. 1-5: Absolute Value Equations and Inequalities • Example: Solve |3x + 6| > 12. Graph the solution.

  23. 1-5: Absolute Value Equations and Inequalities • Example: Solve |3x + 6| > 12. Graph the solution. • Open circle or closed circle? • Come together or split apart?

  24. 1-5: Absolute Value Equations and Inequalities • Example: Solve |3x + 6| > 12. Graph the solution. • Open circle or closed circle? Closed circle (line underneath) • Come together or split apart? Split apart

  25. 1-5: Absolute Value Equations and Inequalities • Solve 3|2x + 6| - 9 < 15. Graph the solution. • Need to get the absolute value alone first. • 3|2x + 6| - 9 < 15 +9 +9 • 3|2x + 6| < 243 3 • |2x + 6| < 8

  26. 1-5: Absolute Value Equations and Inequalities • |2x + 6| < 8

  27. 1-5: Absolute Value Equations and Inequalities • |2x + 6| < 8

  28. 1-5: Absolute Value Equations and Inequalities • |2x + 6| < 8 • Open circle or closed circle? • Come together or split apart?

  29. 1-5: Absolute Value Equations and Inequalities • |2x + 6| < 8 • Open circle or closed circle? Open circle (no line) • Come together or split apart? Come together

  30. 1-5: Absolute Value Equations and Inequalities • Assignment • Page 36 • Problems 16 – 27 (all) • Rest of week, Chapter 1 Test • Wednesday: Preview • Thursday: Review • Friday: Test Day

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