1. Special Equations : AND / OR and Quadratic Inequalities AND / OR are logic operators. AND – where two solution sets “share” common elements. - similar to intersection of two sets OR – where two solution sets are merged together - similar to union of two sets

2. Special Equations : AND / OR and Quadratic Inequalities AND / OR are logic operators. AND – where two solution sets “share” common elements. - similar to intersection of two sets OR – where two solution sets are merged together - similar to union of two sets When utilizing these in graphing multiple inequality equations, a number line graph helps to “see” the final solution.

3. Special Equations : AND / OR and Quadratic Inequalities AND / OR are logic operators. AND – where two solution sets “share” common elements. - similar to intersection of two sets OR – where two solution sets are merged together - similar to union of two sets When utilizing these in graphing multiple inequality equations, a number line graph helps to “see” the final solution. We will first look at how they are different.

4. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE :

5. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other - 3 5

6. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other 2. Graph the solution set for each - 3 5

7. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other 2. Graph the solution set for each 3. Find where they “share” elements - where are they “on top” of each other - in this case they share between (– 3) and 5 - 3 5

8. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other 2. Graph the solution set for each 3. Find where they “share” elements - in this case they share between (– 3) and 5 - where are they “on top” of each other 4. This shared space is our final graph - 3 5

9. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other 2. Graph the solution set for each 3. Find where they “share” elements - in this case they share between (– 3) and 5 - where are they “on top” of each other 4. This shared space is our final graph - 3 5 Answer as an interval

10. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 2 :

11. Special Equations : AND / OR and Quadratic Inequalities STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other - 3 5 EXAMPLE # 2 :

12. Special Equations : AND / OR and Quadratic Inequalities STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other 2. Graph the solution set for each - 3 5 EXAMPLE # 2 :

13. Special Equations : AND / OR and Quadratic Inequalities STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other 2. Graph the solution set for each 3. Find where they “share” elements - where are they “on top” of each other - in this case they share numbers greater than 5 - 3 5 EXAMPLE # 2 :

14. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other 2. Graph the solution set for each 3. Find where they “share” elements - where are they “on top” of each other - in this case they share numbers greater than 5 - 3 5 4. This shared space is our final graph

15. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other 2. Graph the solution set for each 3. Find where they “share” elements - where are they “on top” of each other - in this case they share numbers greater than 5 - 3 5 4. This shared space is our final graph Answer as an interval

16. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 3 :

17. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 3 : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other - 3 5

18. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 3: STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other - 3 5 2. Graph the solution set for each

19. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 3 : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other - 3 5 2. Graph the solution set for each 3. Find where they “share” elements - where are they “on top” of each other - in this case they DO NOT share elements

20. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 3 : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other - 3 5 2. Graph the solution set for each 3. Find where they “share” elements - where are they “on top” of each other - in this case they DO NOT share elements 4. SO we have An EMPTY SET Ø

21. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 4 :

22. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 4 : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other - 3 5

23. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 4 : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other 2. Graph the solution for each - 3 5

24. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 4 : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other 2. Graph the solution for each 3. Now merge the two graphs and keep everything - this will be your answer - 3 5

25. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 4 : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other 2. Graph the solution for each 3. Now merge the two graphs and keep everything - this will be your answer - 3 5 Answer as an interval

26. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 5 : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other 2. Graph the solution for each - 3 5

27. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 5 : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other 2. Graph the solution for each 3. Now merge the two graphs and keep everything - this will be your answer - 3 5

28. Special Equations : AND / OR and Quadratic Inequalities EXAMPLE # 5 : STEPS :1. Create a number line and locate your points. ( open circle for 5 and closed for – 3 ) - when graphing, graph one point higher than the other 2. Graph the solution for each 3. Now merge the two graphs and keep everything - this will be your answer - 3 5 Answer as an interval

29. Special Equations : AND / OR and Quadratic Inequalities Graphing Quadratic Inequalities : 1. Factor to find “critical points” ( where the equation = 0 ) 2. Locate your points on a number line 3. Pick a test point for TRUE or FALSE - true / false changes every time you pass a critical point

30. Special Equations : AND / OR and Quadratic Inequalities Graphing Quadratic Inequalities : 1. Factor to find “critical points” ( where the equation = 0 ) 2. Locate your points on a number line 3. Pick a test point for TRUE or FALSE - true / false changes every time you pass a critical point Example # 1 :Graph the solution set for

31. Special Equations : AND / OR and Quadratic Inequalities Graphing Quadratic Inequalities : 1. Factor to find “critical points” ( where the equation = 0 ) 2. Locate your points on a number line 3. Pick a test point for TRUE or FALSE - true / false changes every time you pass a critical point Example # 1 :Graph the solution set for

32. Special Equations : AND / OR and Quadratic Inequalities Graphing Quadratic Inequalities : 1. Factor to find “critical points” ( where the equation = 0 ) 2. Locate your points on a number line 3. Pick a test point for TRUE or FALSE - true / false changes every time you pass a critical point Example # 1 :Graph the solution set for - 23 Open Circle

33. Special Equations : AND / OR and Quadratic Inequalities Graphing Quadratic Inequalities : 1. Factor to find “critical points” ( where the equation = 0 ) 2. Locate your points on a number line 3. Pick a test point for TRUE or FALSE - true / false changes every time you pass a critical point Example # 1 :Graph the solution set for - 23 Open Circle TTTTTTTTTTTTTTTFFFFFF

34. Special Equations : AND / OR and Quadratic Inequalities Graphing Quadratic Inequalities : 1. Factor to find “critical points” ( where the equation = 0 ) 2. Locate your points on a number line 3. Pick a test point for TRUE or FALSE - true / false changes every time you pass a critical point Example # 1 :Graph the solution set for - 23 Open Circle TTTTTTTTTTTTTTTFFFFFF ALWAYS graph the TRUE sections… Answer as an interval

35. Special Equations : AND / OR and Quadratic Inequalities Graphing Quadratic Inequalities : 1. Factor to find “critical points” ( where the equation = 0 ) 2. Locate your points on a number line 3. Pick a test point for TRUE or FALSE - true / false changes every time you pass a critical point Example # 1 :Graph the solution set for

36. Special Equations : AND / OR and Quadratic Inequalities Graphing Quadratic Inequalities : 1. Factor to find “critical points” ( where the equation = 0 ) 2. Locate your points on a number line 3. Pick a test point for TRUE or FALSE - true / false changes every time you pass a critical point Example # 1 :Graph the solution set for

37. Special Equations : AND / OR and Quadratic Inequalities Graphing Quadratic Inequalities : 1. Factor to find “critical points” ( where the equation = 0 ) 2. Locate your points on a number line 3. Pick a test point for TRUE or FALSE - true / false changes every time you pass a critical point Example # 1 :Graph the solution set for - 4 - 3 Closed Circle

38. Special Equations : AND / OR and Quadratic Inequalities Graphing Quadratic Inequalities : 1. Factor to find “critical points” ( where the equation = 0 ) 2. Locate your points on a number line 3. Pick a test point for TRUE or FALSE - true / false changes every time you pass a critical point Example # 1 :Graph the solution set for - 4 - 3 Closed Circle FFFFF TTTTTTTTTTTTTTTTTTTTTT ALWAYS graph the TRUE sections…

39. Special Equations : AND / OR and Quadratic Inequalities Graphing Quadratic Inequalities : 1. Factor to find “critical points” ( where the equation = 0 ) 2. Locate your points on a number line 3. Pick a test point for TRUE or FALSE - true / false changes every time you pass a critical point Example # 1 :Graph the solution set for - 4 - 3 Closed Circle FFFFF TTTTTTTTTTTTTTTTTTTTTT ALWAYS graph the TRUE sections… Answer as an interval

40. Special Equations : AND / OR and Quadratic Inequalities In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it… Example # 1 :Graph the solution set for 0

41. Special Equations : AND / OR and Quadratic Inequalities In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it… Next, multiply EVERYTHING by “x” and get all terms on one side… Example # 1 :Graph the solution set for 0

42. Special Equations : AND / OR and Quadratic Inequalities In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it… Next, multiply EVERYTHING by “x” and get all terms on one side… Now just follow the steps as we did before… Example # 1 :Graph the solution set for 0-3 6 Closed Circle

43. Special Equations : AND / OR and Quadratic Inequalities In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it… Next, multiply EVERYTHING by “x” and get all terms on one side… Now just follow the steps as we did before… ** I like to test 1000 in these…because 18/1000 is very small, you can disregard it !!! Example # 1 :Graph the solution set for 0-3 6 Closed Circle FTFT

44. Special Equations : AND / OR and Quadratic Inequalities In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it… Next, multiply EVERYTHING by “x” and get all terms on one side… Now just follow the steps as we did before… ** I like to test 1000 in these…because 18/1000 is very small, you can disregard it !!! Example # 1 :Graph the solution set for 0-3 6 Closed Circle FTFT ALWAYS graph the TRUE sections…

45. Special Equations : AND / OR and Quadratic Inequalities In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it… Next, multiply EVERYTHING by “x” and get all terms on one side… Now just follow the steps as we did before… ** I like to test 1000 in these…because 18/1000 is very small, you can disregard it !!! Example # 1 :Graph the solution set for 0-3 6 Closed Circle FTFT ALWAYS graph the TRUE sections… Answer as an interval

46. Special Equations : AND / OR and Quadratic Inequalities In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it… Example # 2 :Graph the solution set for 0

47. Special Equations : AND / OR and Quadratic Inequalities In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it… Next, multiply EVERYTHING by “x” and get all terms on one side… Example # 2 :Graph the solution set for 0

48. Special Equations : AND / OR and Quadratic Inequalities In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it… Next, multiply EVERYTHING by “x” and get all terms on one side… Factor and get critical points… Example # 2 :Graph the solution set for 0 35 Open Circle

49. Special Equations : AND / OR and Quadratic Inequalities In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it… Next, multiply EVERYTHING by “x” and get all terms on one side… Factor and get critical points… Test 1000… Example # 2 :Graph the solution set for 0 35 Open Circle F TT F

50. Special Equations : AND / OR and Quadratic Inequalities In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it… Next, multiply EVERYTHING by “x” and get all terms on one side… Factor and get critical points… Test 1000… Example # 2 :Graph the solution set for 0 35 Open Circle F TT F ALWAYS graph the TRUE sections…

51. Special Equations : AND / OR and Quadratic Inequalities In some cases, the variable will be located in the denominator of a fraction. Since the denominator can not equal zero, zero instantly becomes a critical point with an open circle above it… Next, multiply EVERYTHING by “x” and get all terms on one side… Factor and get critical points… Test 1000… Example # 2 :Graph the solution set for 0 35 Open Circle F TT F ALWAYS graph the TRUE sections… Answer as an interval

52. Special Equations : AND / OR and Quadratic Inequalities An application for using intervals and inequalities is figuring out what you would need on a test to achieve a final grade.

53. Special Equations : AND / OR and Quadratic Inequalities An application for using intervals and inequalities is figuring out what you would need on a test to achieve a final grade. Let’s say you have the following test scores and want a “B” for a final grade. Test scores = 74%, 83%, and a 85%. What range of scores will earn that final grade of a “B” ?

54. Special Equations : AND / OR and Quadratic Inequalities An application for using intervals and inequalities is figuring out what you would need on a test to achieve a final grade. Let’s say you have the following test scores and want a “B” for a final grade. Test scores = 74%, 83%, and a 85%. What range of scores will earn that final grade of a “B” ? First, let’s get a total of your test scores :74 + 83 + 85 = 242

55. Special Equations : AND / OR and Quadratic Inequalities An application for using intervals and inequalities is figuring out what you would need on a test to achieve a final grade. Let’s say you have the following test scores and want a “B” for a final grade. Test scores = 74%, 83%, and a 85%. What range of scores will earn that final grade of a “B” ? First, let’s get a total of your test scores :74 + 83 + 85 = 242 Since there will be four tests used to calculate the final grade, 400 points is the total possible points that could be earned.

56. Special Equations : AND / OR and Quadratic Inequalities An application for using intervals and inequalities is figuring out what you would need on a test to achieve a final grade. Let’s say you have the following test scores and want a “B” for a final grade. Test scores = 74%, 83%, and a 85%. What range of scores will earn that final grade of a “B” ? First, let’s get a total of your test scores :74 + 83 + 85 = 242 Since there will be four tests used to calculate the final grade, 400 points is the total possible points that could be earned.

57. Special Equations : AND / OR and Quadratic Inequalities An application for using intervals and inequalities is figuring out what you would need on a test to achieve a final grade. Let’s say you have the following test scores and want a “B” for a final grade. Test scores = 74%, 83%, and a 85%. What range of scores will earn that final grade of a “B” ? First, let’s get a total of your test scores :74 + 83 + 85 = 242 Since there will be four tests used to calculate the final grade, 400 points is the total possible points that could be earned. So far, this is the equation we have adding that final test and the possible points :

58. Special Equations : AND / OR and Quadratic Inequalities An application for using intervals and inequalities is figuring out what you would need on a test to achieve a final grade. Let’s say you have the following test scores and want a “B” for a final grade. Test scores = 74%, 83%, and a 85%. What range of scores will earn that final grade of a “B” ? First, let’s get a total of your test scores :74 + 83 + 85 = 242 Since there will be four tests used to calculate the final grade, 400 points is the total possible points that could be earned. So far, this is the equation we have adding that final test and the possible points : If we need ATLEAST an 80% for a “B”, then the equation needs to be greater than or equal to 80% or 0.80

59. Special Equations : AND / OR and Quadratic Inequalities An application for using intervals and inequalities is figuring out what you would need on a test to achieve a final grade. Let’s say you have the following test scores and want a “B” for a final grade. Test scores = 74%, 83%, and a 85%. What range of scores will earn that final grade of a “B” ? First, let’s get a total of your test scores :74 + 83 + 85 = 242 Since there will be four tests used to calculate the final grade, 400 points is the total possible points that could be earned. So far, this is the equation we have adding that final test and the possible points : Now let’s solve for “x”

60. Special Equations : AND / OR and Quadratic Inequalities An application for using intervals and inequalities is figuring out what you would need on a test to achieve a final grade. Let’s say you have the following test scores and want a “B” for a final grade. Test scores = 74%, 83%, and a 85%. What range of scores will earn that final grade of a “B” ? First, let’s get a total of your test scores :74 + 83 + 85 = 242 Since there will be four tests used to calculate the final grade, 400 points is the total possible points that could be earned. So far, this is the equation we have adding that final test and the possible points : Now let’s solve for “x” Multiplied both sides by 400

61. Special Equations : AND / OR and Quadratic Inequalities An application for using intervals and inequalities is figuring out what you would need on a test to achieve a final grade. Let’s say you have the following test scores and want a “B” for a final grade. Test scores = 74%, 83%, and a 85%. What range of scores will earn that final grade of a “B” ? First, let’s get a total of your test scores :74 + 83 + 85 = 242 Since there will be four tests used to calculate the final grade, 400 points is the total possible points that could be earned. So far, this is the equation we have adding that final test and the possible points : Now let’s solve for “x” Multiplied both sides by 400 Subtracted both sides by 242

62. Special Equations : AND / OR and Quadratic Inequalities An application for using intervals and inequalities is figuring out what you would need on a test to achieve a final grade. Let’s say you have the following test scores and want a “B” for a final grade. Test scores = 74%, 83%, and a 85%. What range of scores will earn that final grade of a “B” ? First, let’s get a total of your test scores :74 + 83 + 85 = 242 Since there will be four tests used to calculate the final grade, 400 points is the total possible points that could be earned. So far, this is the equation we have adding that final test and the possible points : Now let’s solve for “x” Multiplied both sides by 400 Subtracted both sides by 242 So the range of scores for a “B” = [ 78,100 ]