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 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:
The upper limit is 9. Because x can be equal to 9, we mark it with a filled-in circle. 5 10 15
-20
-15
-10 -5
-25 20 25

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

32. To graph a compound inequality:
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. ? ? ? 5 10 15
-20
-15
-10 -5
-25 20 25

33. 1. Find and mark the boundary points.
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 5 10 15
-20
-15
-10 -5
-25 20 25

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

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

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

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

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

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

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

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

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

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

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

45. So it would be incorrect to use and
So it would be incorrect to use and. And implies that a number meets both conditions. 5 10 15
-20
-15
-10 -5
-25 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:5 10 15
-20
-15
-10 -5
-25 20 25
3: Write variable:
1: Write boundaries:
2: Write signs:

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

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