2. You already know inequalities. Often they are used to place limits on variables. That just means x can be any number equal to 9 or less than 9.

3. Sometimes we put more than one limit on the variable: Now x is still less than or equal to 9, but it must also be greater than or equal to –7.

4. Let’s look at the graph: 051015-20-15-10-5-252025 The upper limit is 9. Because x can be equal to 9, we mark it with a filled-in circle.

5. 051015-20-15-10-5-252025 The lower limit is -7. We also need to mark it with a filled-in circle.

6. Where are they found on the graph? Yes! It is less than or equal to 9? What about –15? 051015-20-15-10-5-252025 There are other numbers that satisfy both conditions.

7. It is also greater than or equal to -7? What about –15? 051015-20-15-10-5-252025 No! Where are they found on the graph?

8. Because the word and is used, a number on the graph needs to satisfy both parts of the inequality. 051015-20-15-10-5-252025

9. Yes! So let’s try 20. Does 20 satisfy both conditions? 051015-20-15-10-5-252025

10. No! So let’s try 20. Does 20 satisfy both conditions? 051015-20-15-10-5-252025

11. Since 20 does not satisfy both conditions, it can’t belong to the solution set. 051015-20-15-10-5-252025

12. There is one region we have not checked. 051015-20-15-10-5-252025

13. We need to choose a number from that region. 051015-20-15-10-5-252025 You want to choose 0? Good choice! 0 is usually the easiest number to work with.

14. Does 0 satisfy both conditions? 051015-20-15-10-5-252025 Yes!

15. Does 0 satisfy both conditions? 051015-20-15-10-5-252025 Yes!

16. If one number in a region completely satisfies an inequality, 051015-20-15-10-5-252025 you can know that every number in that region satisfies the inequality.

17. And Shade between the two numbers. Make sure you think if it makes sense because no solutions can exist. 051015-20-15-10-5-252025

18. Let’s graph another inequality: 051015-20-15-10-5-252025

19. tells us we want an open circle, The first sign First we mark the boundary points: 051015-20-15-10-5-252025

20. and the 12 tells us where the circle goes. 051015-20-15-10-5-252025

21. and the 12 tells us where the circle goes. 051015-20-15-10-5-252025

22. tells us we want a closed circle, The second sign 051015-20-15-10-5-252025

23. and the -1 tells us where the circle goes. 051015-20-15-10-5-252025

24. The boundary points divide the line into three regions: 051015-20-15-10-5-252025 123

25. We need to test one point from each region. 051015-20-15-10-5-252025 No!Yes!

26. Notice that the word used is or, 051015-20-15-10-5-252025 No!Yes! instead of and.

27. Or means that a number 051015-20-15-10-5-252025 No!Yes! only needs to meet one condition.

28. Because –10 meets one condition, 051015-20-15-10-5-252025 Yes! the region to which it belongs...... belongs to the graph.

29. Let’s check the next region: 051015-20-15-10-5-252025 No!

30. Because –1 meets neither condition, 051015-20-15-10-5-252025 No! the numbers in that region will not satisfy the inequality.

31. Now the final region: 051015-20-15-10-5-252025 Yes!No!

32. Again, 15 meets one condition 051015-20-15-10-5-252025 Yes! so we need to shade that region.

33. Or shade outside the boundaries 051015-20-15-10-5-252025 Make sure you think because no solutions do exist

34. A quick review: 051015-20-15-10-5-252025 1. Find and mark the boundary points. 2. Test points from each region. 3. Shade the regions that satisfy the inequality. orandor To graph a compound inequality:

35. A quick review: 051015-20-15-10-5-252025 1. Find and mark the boundary points. 2. Test points from each region. 3. Shade the regions that satisfy the inequality. or

36. Given the graph below, write the inequality. First, write the boundary points. 051015-20-15-10-5-252025

37. Then look at the marks on the graph, 051015-20-15-10-5-252025 and write the correct symbol.

38. Since x is between the boundary points on the graph, 051015-20-15-10-5-252025 it will be between the boundary points in the inequality.

39. Since x is between the boundary points on the graph, 051015-20-15-10-5-252025 it will be between the boundary points in the inequality.

40. 051015-20-15-10-5-252025 Try this one:

41. 051015-20-15-10-5-252025 And again, you need to choose the correct symbols:

42. 051015-20-15-10-5-252025 Because the x -values are not between the boundary points on the graph, we won’t write x between the boundary points in the equation.

43. 051015-20-15-10-5-252025 Because the x -values are not between the boundary points on the graph, we won’t write them between the boundary points in the equation.

44. 051015-20-15-10-5-252025 We will use the word, or, instead: Remember that or means a number has to satisfy only one of the conditions.

45. 051015-20-15-10-5-252025 We will use the word, or, instead: Remember that or means a number has to satisfy only one of the conditions.

46. 051015-20-15-10-5-252025 Is there any one number that belongs to both shaded sections in the graph? Say NO! NO!

47. 051015-20-15-10-5-252025 So it would be incorrect to use and. And implies that a number meets both conditions.

48. ... you remember that a compound inequality is just two inequalities put together. Solving compound inequalities is easy if...

49. You can solve them both at the same time:

50. Write the inequality from the graph: 051015-20-15-10-5-252025 1:Write boundaries:2:Write signs:3:Write variable:

51. Solve the inequality: Is this what you did?

52. You did remember to reverse the signs...... didn’t you?Good job!

53. HW: pg. 585 #2 to 40 even