1.4 Absolute Values

1. 1.4 Absolute Values Solving Absolute Value Equations By putting into one of 3 categories

2. What is the definition of “Absolute Value”? __________________________________ __________________________________ Mathematically, ___________________ For example, in , where could x be? 0 3 -3

3. To solve these situations, ______________________________________________________________________________________________________ Consider ________________________ 0

4. That was Category 1 Category 1 : ____________________________ _______________________________________ _______________________________________ Example Case 1 Case 2

5. Absolute Value Inequalities Think logically about another situation. What does mean? For instance, in the equation , _________________________________ 0

6. How does that translate into a sentence? __________________________________ Now solve for x. This is Category 2: when x is less than a number

7. Absolute Value Inequalities What does mean? In the equation , __________________________________ 0

8. Less than = And statement Greater than = Or statement Note:  is the same as ; is the same as ; just have the sign in the rewritten equation match the original.

9. Isolate Absolute Value • _______________________________________ • ______________________________________ • ______________________________________

10. When x is on both sides • ______________________________________ ________________________________________ • ______________________________________ Case 1 Case 2

11. Example inequality with x on other side Case 1 Case 2

12. Examples Case 1 Case 2

13. Case 1 Case 2

14. Case 1 Case 2

15. _________________________________ Whats wrong with this? ____________________________________________________________________________________

17. 1-4 Compound Absolute ValuesEqualities and Inequalities More than one absolute value in the equation

18. Some Vocab • Domain- _________________________ • Range- __________________________ • Restriction- _______________________ • __________________________________________________________________

19. Find a number that works. We will find a more methodical approach to find all the solutions.

20. In your approach, think about the values of this particular mathematical statement in the 3 different areas on a number line. • -_________________________________________ • ________________________________________ • - ________________________________________

21. It now forms 3 different areas or cases on the number line. • ______________________________ • ______________________________ • ___________________________________

22. Case 1 Domain ________________________ ____________________________________________________ ___________________________________________________________________________________________________

23. Case 2 Domain ___________________________________

24. Case 3 Domain _____________________________

25. Final Answers • ______________________________ • ______________________________ • _____________________________ 2 5 -2

26. If you get an answer in any of the cases where the variable disappears and the answer is TRUE ________________________________ ________________________________________________________________________________________________

28. 1-5 Exponential Rules You know these already

29. Review of the Basics

30. Practice problems Simplify each expression

32. Do NOT use calculators on the homework, please!! 

33. 1-6 Radicals (Day 1) and Rational Exponents (Day 2)

34. What is a Radical? In simplest term, it is a square root ( ) = ________________

35. The Principle nth root of a: 1. • If a < 0 and n is odd then is a negative number ___________________ • If a < 0 and n is even, _____________ __________

36. Vocab

37. Lets Recall

38. Rules of Radicals

39. Practice problems Simplify each expression

40. 1-6 Radicals (Day 1) and Rational Exponents (Day 2)

41. What is a Rational Exponent? ____________________________________ ____________________________________ ____________________________________ ____________________________________

42. Don’t be overwhelmed by fractions! These problems are not hard, as long as you remember what each letter means. Notice I used “p” as the numerator and “r” as the denominator. p= _____________________________ r= _____________________________ ________________________________________ ________________________________________ ________________________________________

43. Practice problems Simplify each expression

44. The last part of this topic What is wrong with this number? ___________________________________ ___________________________________ ___________________________________

45. Rationalizing If you see a single radical in the denominator ______________________________________ ________________________________________ ________________________________________ ________________________________________

46. Rationalizing If you see 1 or more radicals in a binomial, what can we do?

47. What is a Conjugate? The conjugate of a + b ______. Why? The conjugate _______________________ ___________________________________ ___________________________________ ___________________________________

48. Rationalizing Multiply top and bottom by the conjugate! Its MAGIC!!

50. 1.7 Fundamental Operations