Compound

1. Compound Inequalities

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 isstill less than or equal to 9, but it must also be greater than or equal to –7.

4. -25 -20 -15 -10 -5 0 5 10 15 20 25 Let’s look at the graph: The upper limit is 9. Because xcan be equal to 9, we mark it with a filled-in circle.

5. -25 -20 -15 -10 -5 0 5 10 15 20 25 The lower limit is -7. We also need to mark it with a filled-in circle.

6. There are other numbers that satisfy both conditions. -25 -20 -15 -10 -5 0 5 10 15 20 25 Where are they found on the graph? What about –15? It is less than or equal to 9? Yes!

7. -25 -20 -15 -10 -5 0 5 10 15 20 25 Where are they found on the graph? What about –15? It is also greater than or equal to -7? No!

8. Because the word and is used, a number on the graph needs to satisfy both parts of the inequality. -25 -20 -15 -10 -5 0 5 10 15 20 25

9. So let’s try 20. Does 20 satisfy both conditions? -25 -20 -15 -10 -5 0 5 10 15 20 25 Yes!

10. So let’s try 20. Does 20 satisfy both conditions? -25 -20 -15 -10 -5 0 5 10 15 20 25 No!

11. Since 20 does not satisfy both conditions, it can’t belong to the solution set. -25 -20 -15 -10 -5 0 5 10 15 20 25

12. There is one region we have not checked. -25 -20 -15 -10 -5 0 5 10 15 20 25

13. We need to choose a number from that region. -25 -20 -15 -10 -5 0 5 10 15 20 25 You want to choose 0? Good choice! 0 is usually the easiest number to work with.

14. Does 0 satisfy both conditions? -25 -20 -15 -10 -5 0 5 10 15 20 25 Yes!

15. Does 0 satisfy both conditions? -25 -20 -15 -10 -5 0 5 10 15 20 25 Yes!

16. If one number in a region completely satisfies an inequality, -25 -20 -15 -10 -5 0 5 10 15 20 25 you can know that every number in that region satisfies the inequality.

17. Let’s graph another inequality: -25 -20 -15 -10 -5 0 5 10 15 20 25

18. First we mark the boundary points: -25 -20 -15 -10 -5 0 5 10 15 20 25 The first sign tells us we want an open circle,

19. -25 -20 -15 -10 -5 0 5 10 15 20 25 and the 12 tells us where the circle goes.

20. -25 -20 -15 -10 -5 0 5 10 15 20 25 and the 12 tells us where the circle goes.

21. -25 -20 -15 -10 -5 0 5 10 15 20 25 The second sign tells us we want a closed circle,

22. -25 -20 -15 -10 -5 0 5 10 15 20 25 and the -1 tells us where the circle goes.

23. -25 -20 -15 -10 -5 0 5 10 15 20 25 The boundary points divide the line into three regions: 1 2 3

24. -25 -20 -15 -10 -5 0 5 10 15 20 25 We need to test one point from each region. No! Yes!

25. -25 -20 -15 -10 -5 0 5 10 15 20 25 Notice that the word used is or, instead of and. No! Yes!

26. -25 -20 -15 -10 -5 0 5 10 15 20 25 Or means that a number only needs to meet one condition. No! Yes!

27. -25 -20 -15 -10 -5 0 5 10 15 20 25 Because –10 meets one condition, the region to which it belongs . . . . . . belongs to the graph. Yes!

28. -25 -20 -15 -10 -5 0 5 10 15 20 25 Let’s check the next region: No! No!

29. No! No! -25 -20 -15 -10 -5 0 5 10 15 20 25 Because –1 meets neither condition, the numbers in that region will not satisfy the inequality.

30. -25 -20 -15 -10 -5 0 5 10 15 20 25 Now the final region: Yes! No!

31. -25 -20 -15 -10 -5 0 5 10 15 20 25 Again, 15 meets one condition so we need to shade that region. Yes!

32. -25 -20 -15 -10 -5 0 5 10 15 20 25 A quick review: To graph a compound inequality: 1. Find and mark the boundary points. 2. Test points from each region. 3. Shade the regions that satisfy the inequality. ? ? ?

33. -25 -20 -15 -10 -5 0 5 10 15 20 25 A quick review: 1. Find and mark the boundary points. 2. Test points from each region. 3. Shade the regions that satisfy the inequality. or

34. -25 -20 -15 -10 -5 0 5 10 15 20 25 Given the graph below, write the inequality. First, write the boundary points.

35. -25 -20 -15 -10 -5 0 5 10 15 20 25 Then look at the marks on the graph, and write the correct symbol.

36. -25 -20 -15 -10 -5 0 5 10 15 20 25 Since x is between the boundary points on the graph, it will be between the boundary points in the inequality.

37. -25 -20 -15 -10 -5 0 5 10 15 20 25 Since x is between the boundary points on the graph, it will be between the boundary points in the inequality.

38. -25 -20 -15 -10 -5 0 5 10 15 20 25 Try this one: Again, begin by writing the boundary points:

39. -25 -20 -15 -10 -5 0 5 10 15 20 25 And again, you need to choose the correct symbols:

40. -25 -20 -15 -10 -5 0 5 10 15 20 25 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.

41. -25 -20 -15 -10 -5 0 5 10 15 20 25 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.

42. -25 -20 -15 -10 -5 0 5 10 15 20 25 We will use the word, or, instead: Remember that or means a number has to satisfy only one of the conditions.

43. -25 -20 -15 -10 -5 0 5 10 15 20 25 We will use the word, or, instead: Remember that or means a number has to satisfy only one of the conditions.

44. -25 -20 -15 -10 -5 0 5 10 15 20 25 Is there any one number that belongs to both shaded sections in the graph? NO! Say NO!

45. So it would be incorrect to use and. And implies that a number meets both conditions. -25 -20 -15 -10 -5 0 5 10 15 20 25

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

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

48. Write the inequality from the graph: -25 -20 -15 -10 -5 0 5 10 15 20 25 1: Write boundaries: 2: Write signs: 3: Write variable:

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

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