The document discusses various methods for solving inequalities, including: - Properties for adding, subtracting, multiplying, and dividing terms within an inequality - Using set-builder and interval notation to describe the solution set of an inequality - Graphical representations using open and closed circles to indicate whether a number is or isn't part of the solution set The document provides examples of applying these different techniques to solve specific inequalities.

1. Solving Inequalities

3. For any two real numbers, a and b , exactly one of the following statements is true . Solving Inequalities

4. For any two real numbers, a and b , exactly one of the following statements is true . Solving Inequalities

5. For any two real numbers, a and b , exactly one of the following statements is true . Solving Inequalities

6. For any two real numbers, a and b , exactly one of the following statements is true . Solving Inequalities

7. For any two real numbers, a and b , exactly one of the following statements is true . This is known as the Trichotomy Property Solving Inequalities

8. For any two real numbers, a and b , exactly one of the following statements is true . This is known as the Trichotomy Property or the property of order . Solving Inequalities

9. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Solving Inequalities

10. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities

11. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b

12. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c c

13. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c c

14. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c

15. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c

16. Adding the same number to, or subtracting the same number from, each side of an inequality does not change the truth of the inequality. Addition Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b a+c b+c

17. Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities

18. Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a b

19. Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b c

20. Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b c

21. Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b

22. Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b

23. Subtraction Property of Inequality For any real numbers a , b , and c : Solving Inequalities a-c b-c a b

24. Solving Inequalities a

25. Solving Inequalities a

26. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a

27. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a a

28. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a a

29. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. We use a closed circle (dot) to indicate that a IS part of the solution set. a a

30. Solving Inequalities a

31. Solving Inequalities a

32. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a

33. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a a

34. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. a a

35. Solving Inequalities We use an open circle (dot) to indicate that a is NOT part of the solution set. We use a closed circle (dot) to indicate that a IS part of the solution set. a a

36. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Solving Inequalities

37. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities

38. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities

39. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive:

40. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive:

41. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive:

42. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive: c is negative:

43. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive: c is negative:

44. Multiplying or dividing each side of an inequality by a positive number does not change the truth of the inequality. Multiplication Property of Inequality For any real numbers a , b , and c : However , multiplying or dividing each side of an inequality by a negative number requires that the order of the inequality be reversed . Solving Inequalities c is positive: c is negative:

45. Division Property of Inequality Most books run us through the “rules” for division. Why is this not necessary? Solving Inequalities

46. Division Property of Inequality Most books run us through the “rules” for division. Why is this not necessary? Solving Inequalities HINT: is the same as

47. Division Property of Inequality Most books run us through the “rules” for division. Why is this not necessary? Solving Inequalities HINT: is the same as

48. Division Property of Inequality Most books run us through the “rules” for division. Why is this not necessary? So, see rules for multiplication! Solving Inequalities HINT: is the same as

49. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities 4

50. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities 4 set-builder notation

51. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is less than 4 } 4 set-builder notation

52. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is less than 4 } 4 set-builder notation Identify the variable used

53. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is less than 4 } 4 set-builder notation Identify the variable used Describe the limitations or boundary of the variable

54. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities -7

55. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities -7 set-builder notation

56. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is greater than or equal to negative 7 } -7 set-builder notation

57. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is greater than or equal to negative 7 } Identify the variable used -7 set-builder notation

58. The solution set of an inequality can also be described by using set-builder notation . Solving Inequalities Read: { x “such that” x is greater than or equal to negative 7 } Identify the variable used Describe the limitations or boundary of the variable -7 set-builder notation

59. The solution set of an inequality can also be described by using interval notation . Solving Inequalities

60. The solution set of an inequality can also be described by using interval notation . Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively.

61. The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively.

62. The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4

63. The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4 interval notation

64. The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. To indicate that an endpoint is included in the solution set, a bracket, [ or ], is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4 interval notation

65. The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. To indicate that an endpoint is included in the solution set, a bracket, [ or ], is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4 interval notation -7

66. The solution set of an inequality can also be described by using interval notation . To indicate that an endpoint is not included in the solution set, a parenthesis, ( or ), is used. To indicate that an endpoint is included in the solution set, a bracket, [ or ], is used. Solving Inequalities The infinity symbols and are used to indicate that a set is unbounded in the positive or negative direction, respectively. 4 interval notation -7 interval notation

67. End of Lesson Solving Inequalities

68. Credits PowerPoint created by Using Glencoe’s Algebra 2 text, © 2005 Robert Fant http://robertfant.com