1. Ultimatealgebra.com

2. WELCOME TO THIS CHAPTER 12 part 1 WE WOULD BE LEARNING ABOUT INEQUALITIES PLEASE MASTER THESE BEFORE YOU MOVE TO THE NEXT PART BECAUSE IF YOU CAN’T WORK WITH INEQUALITIES YOU WOULD UNDERSTAND THE REST OF THE COURSE

3. IN THIS CHAPTER WE WILL LEARN INTRODUCTION TO INEQUALITIES SIGNS OR SYMBOLS IN INEQUALITIES COMBINING INEQUALITIES

4. In algebra, in math, and in life, things are not always equal. In your house you might be older or younger than your siblings. Such situations brings about the necessity of the introduction of inequalities EXAMPLE

5. Also if you are younger than your brother we say your age is less than your brother’s age This is represented as Your age < brother’s age So when you are older than your brother we say your age is greater than your brother’s age This is represented as Your age > brother’s age

6. There are times where things are not exactly greater than or less than but are from a point up or from a point down we might want to say all our friends are thirteen years or older. We cannot use just the greater than sign here because it is possible some of your friends are exactly thirteen years EXAMPLE

7. In exactly the same way we might want to say I have no friend whose age is above 20. By saying this you are implying that your friends’ ages can be 20 or less.

8. REVISION OF SIGNS

9. Solving linear inequalities EXAMPLE The process of solving inequality is exactly like solving equalities. The sign is the only difference

10. Solving linear inequalities EXAMPLE There is a little difference when it comes to dividing or multiplying by a negative numbers To get your final value you have to divide by -2

11. Solving linear inequalities When you divide or multiply by a negative number the inequality sign changes Sign change. Sign change. Sign change. sign Notice how the sign changed from > to < as we divided by -2

12. MORE ON EXPRESSING RANGE There are times that we want to say things in a range. We want to say from one point to the other 1. My test score ranges from 100% to 85% EXAMPLES 2. All my books are either below 20 pages or above 100 pages 3. I bought my used books not more than $60 and my new books not less than $100

13. MORE ON EXPRESSING RANGE 1. My test score ranges from 100% to 85% Let my test score = x My test score ranging from 100% to 85% means it was less than or equal to 100% and greater than or equal to 85%

14. MORE ON EXPRESSING RANGE What you should pay close attention to is where the open and closed part of the inequality sign faces in relation to the variable and the number and make sure that after combining it, it has the same effect

15. MORE ON EXPRESSING RANGE 2. All my books are either below 20 pages or above 100 pages Let x = number of pages Here we are saying that the pages of the books are less than 20 pages (no book is exactly 20 pages). So we use just the less than sign Again we are saying that the pages of the books can also be greater than 100 pages (no book is exactly 100 pages). So we use just the greater than sign

16. MORE ON EXPRESSING RANGE Yes it makes sense if you consider the explanation in example 1. But you cannot join two inequalities if their ranges do not over lap

17. Why we can put some inequalities together but not others In the first example, we talk about an x value which is between 100% and 85%.

18. Why we can put some inequalities together but not others

19. So we can understand that we can only combine two inequalities only if any value of x chosen satisfies both equations.

20. IN THIS CHAPTER WE LEARNT INTRODUCTION TO INEQUALITIES SIGNS OR SYMBOLS IN INEQUALITIES COMBINING INEQUALITIES Ultimatealgebra.com

23. WELCOME THIS CHAPTER WE WOULD BE LEARNING GRAPHING OF INQUALITIES IN ONE VARIABLE PLEASE MASTER THESE BEFORE YOU MOVE TO NEXT CHAPTER BECAUSE IF YOU CAN’T WORK WITH INEQUALITIES YOU WOULD UNDERSTAND THE REST OF THE COURSE

24. IN THIS CHAPTER WE WILL LEARN GRAPHING ON THE NUMBER LINE SYMBOLS FOR GRAPHING CHEAT METHOD TO GRAPHING GRAPHING OF RANGE

25. Graphing on the number line

26. EXAMPLE 1

27. Let’s cheat If you can write your formula in the form below then your arrow would always points just like the sign you have X > 1 variable, inequality symbol then number

28. EXAMPLE 2

29. EXAMPLE 3

30. EXAMPLE 4 Here in order to use the cheat method you should have variable, inequality sign and then number. But this is not like that so we have to convert. What you do is kind of rotate the question.

31. EXAMPLE 1 MORE ON THE NUMBER LINE OF RANGE My test score ranges from 100% to 85%. Plot on the number line For the purpose of plotting on the number line it is easier to keep equations without combining them. You can combine them using the information you get from the graph

32. MORE ON THE NUMBER LINE OF RANGE

33. Let’s simplify the steps 1. Find your two points ( in this case 85 and 100) 2. Mark them with a shaded or not shaded circles ( in this case shaded because it has equal signs) 3. Draw extended arrows, if the two arrows overlap, clean the excess and that would be your answer.

34. All my books are either below 20 pages or above 100 pages EXAMPLE 2

35. MORE ON THE NUMBER LINE OF RANGE Let’s simplify the steps 1. Find your two points ( in this case 20 and 100) 2. Mark them with a shaded or not shaded circles ( in this case not shaded because it has no equal signs) 3. Draw extended arrows; if the two arrows do not overlap you are done with your answer.

36. Breaking out two inequalities We already know how to put two inequalities together. It would be easy to just leave you to figure out how to reverse the process but I would like to give some examples to explain this. 1. Break 2<x<4 into two separate inequalities x>2 and x<4 EXAMPLE Break 3>x<5 into separate inequalities x<3 and x<5

37. IN THIS CHAPTER WE LEARNT GRAPHING ON THE NUMBER LINE SYMBOLS FOR GRAPHING CHEAT METHOD TO GRAPHING GRAPHING OF RANGE Ultimatealgebra.com

40. WELCOME THIS CHAPTER WE WOULD BE LEARNING GRAPHING OF INQUALITIES IN ONE VARIABLE PLEASE MASTER THESE BEFORE YOU MOVE TO NEXT CHAPTER BECAUSE IF YOU CAN’T WORK WITH INEQUALITIES YOU WOULD UNDERSTAND THE REST OF THE COURSE

41. IN THIS CHAPTER WE WILL LEARN GRAPHING ON THE NUMBER LINE SYMBOLS FOR GRAPHING CHEAT METHOD TO GRAPHING GRAPHING OF RANGE

42. Graphing on the number line

43. EXAMPLE 1

44. Let’s cheat If you can write your formula in the form below then your arrow would always points just like the sign you have X > 1 variable, inequality symbol then number

45. EXAMPLE 2

46. EXAMPLE 3

47. EXAMPLE 4 Here in order to use the cheat method you should have variable, inequality sign and then number. But this is not like that so we have to convert. What you do is kind of rotate the question.

48. EXAMPLE 1 MORE ON THE NUMBER LINE OF RANGE My test score ranges from 100% to 85%. Plot on the number line For the purpose of plotting on the number line it is easier to keep equations without combining them. You can combine them using the information you get from the graph

49. MORE ON THE NUMBER LINE OF RANGE

50. Let’s simplify the steps 1. Find your two points ( in this case 85 and 100) 2. Mark them with a shaded or not shaded circles ( in this case shaded because it has equal signs) 3. Draw extended arrows, if the two arrows overlap, clean the excess and that would be your answer.

51. All my books are either below 20 pages or above 100 pages EXAMPLE 2

52. MORE ON THE NUMBER LINE OF RANGE Let’s simplify the steps 1. Find your two points ( in this case 20 and 100) 2. Mark them with a shaded or not shaded circles ( in this case not shaded because it has no equal signs) 3. Draw extended arrows; if the two arrows do not overlap you are done with your answer.

53. Breaking out two inequalities We already know how to put two inequalities together. It would be easy to just leave you to figure out how to reverse the process but I would like to give some examples to explain this. 1. Break 2<x<4 into two separate inequalities x>2 and x<4 EXAMPLE Break 3>x<5 into separate inequalities x<3 and x<5

54. IN THIS CHAPTER WE LEARNT GRAPHING ON THE NUMBER LINE SYMBOLS FOR GRAPHING CHEAT METHOD TO GRAPHING GRAPHING OF RANGE Ultimatealgebra.com