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Compound

Compound. Inequalities. 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. Sometimes we put more than one limit on the variable:.

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Compound

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  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. And Shade between the two numbers. Make sure you think if it makes sense because no solutions can exist. -25 -20 -15 -10 -5 0 5 10 15 20 25

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

  19. 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,

  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 and the 12 tells us where the circle goes.

  22. -25 -20 -15 -10 -5 0 5 10 15 20 25 The second sign tells us we want a closed circle,

  23. -25 -20 -15 -10 -5 0 5 10 15 20 25 and the -1 tells us where the circle goes.

  24. -25 -20 -15 -10 -5 0 5 10 15 20 25 The boundary points divide the line into three regions: 1 2 3

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

  26. -25 -20 -15 -10 -5 0 5 10 15 20 25 Notice that the word used is or, instead of and. No! Yes!

  27. -25 -20 -15 -10 -5 0 5 10 15 20 25 Or means that a number only needs to meet one condition. No! Yes!

  28. -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!

  29. -25 -20 -15 -10 -5 0 5 10 15 20 25 Let’s check the next region: No! No!

  30. 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.

  31. -25 -20 -15 -10 -5 0 5 10 15 20 25 Now the final region: Yes! No!

  32. -25 -20 -15 -10 -5 0 5 10 15 20 25 Again, 15 meets one condition so we need to shade that region. Yes!

  33. -25 -20 -15 -10 -5 0 5 10 15 20 25 Or shade outside the boundaries Make sure you think because no solutions do exist

  34. -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. or and or

  35. -25 -20 -15 -10 -5 0 5 10 15 20 25 Given the graph below, write the inequality. First, write the boundary points.

  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:

  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 We will use the word, or, instead: Remember that or means a number has to satisfy only one of the conditions.

  42. -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!

  43. 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

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

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

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

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

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

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