1. Lesson 4.1
Inequalities pp

2. Objectives:
1. To identify the addition, multiplication, and transitive properties of inequalities.
2. To apply the definition and properties of inequalities to graphing.

3. Inequality Properties
Addition property: If a > b, the a + c > b + c.
Multiplication property: If a > b and c > 0, then ac > bc If a > b and c < 0, then ac < bc.

4. Inequality Properties
Transitive property: If a > b and b > c, then a > c.

5. Practice: If a < b and c = -5, then which of the following is true?
1. ac < bc
2. ac > bc
3. ac = bc

6. Graph x  3 or x < 0. 3

7. Graph x  3 or x < 0. 3

8. Graph x  3 or x < 0. 3

9. Graph x  3 and x < 0. 3 Ø

10. 1. For |x| < a, x is between -a and a -a < x < a
To graph absolute value inequalities
1. For |x| < a, x is between
-a and a
-a < x < a
2. For |x| > a, x is greater
than a or x is less than -a
x < -a or x > a
3. For |x| = a, x = a or
x = -a

11. Example 3: Graph |x| < 5.
|x| < 5 means that x is less than 5 units from the origin.
We write -5 < x and x < 5, which is -5 < x < 5. 3 6 -3 -6

12. Graph |x| > 5.
|x| > 5 means that x is more that 5 units from the origin.
We write -5 > x or x > 5, which is x < -5 or x > 5. 3 6 -3 -6

13. Graph |x| < -5.
Is it possible for the absolute value of a number to be negative? Ø 3 6 -3 -6

14. Definition
A real number a is greater than a real number b (a > b) if there is a positive real number c so that a = b + c.

15. Graph x  5 and x < -2. 1. 2. 3. 3 6 -3 -6

16. Graph x  5 or x < -2. 1. 2. 3. 3 6 -3 -6

17. Graph x  5 and x > -2. 1. 2. 3. 3 6 -3 -6

18. Graph x  5 or x > -2. 1. 2. 3. 3 6 -3 -6

19. Solve the inequality |x| > 4.
3. x < -4 or x > 4
4. -4 < x < 4

20. Graph |x| > 4. 1. 2. 3. 3 6 -3 -6

21. Graph |x-3| < 5. 1. 2. 3. 3 6 9 -3 -6

22. 1. Match each compound inequality with its graph. x  -2 and x  1
1. 2.
-2 0 2
-2 0 2
3. 4. Ø
-2 0 2

23. 2. Match each compound inequality with its graph. x  -2 or x  1
1. 2.
-2 0 2
-2 0 2
3. 4. Ø
-2 0 2

24. 3. Match each compound inequality with its graph. x  -2 and x  1
1. 2.
-2 0 2
-2 0 2
3. 4.
-2 0 2
-2 0 2

25. 4. Match each compound inequality with its graph. x  -2 or x  1
1. 2.
-2 0 2
-2 0 2
3. 4.
-2 0 2
-2 0 2

26. 5. Match each absolute value inequality with its graph. |x|  2
1. 2.
-2 0 2
-2 0 2
3. 4.
-2 0 2
-2 0 2

27. 6. Match each absolute value inequality with its graph. |x|  2
1. 2.
-2 0 2
-2 0 2
3. 4.
-2 0 2
-2 0 2

28. 7. Match each absolute value inequality with its graph. |x|  -2
1. 2.
-2 0 2
-2 0 2
3. 4. Ø
-2 0 2

29. 8. Match each absolute value inequality with its graph. |x|  -2
1. 2.
-2 0 2
-2 0 2
3. 4. Ø
-2 0 2

30. Homework pp

31. ►A. Exercises
Identify each property.
1. x  2 and 2  5, therefore x  5.
1. Addition
2. Multiplication
3. Transitive
4. Def. of greater than

32. ►A. Exercises
Identify each property.
3. Since x  5, x = 5 + c for some constant c.
1. Addition
2. Multiplication
3. Transitive
4. Def. of greater than

33. ►A. Exercises
Graph.
5. x  2 and x  1 3 6 -3 -6

34. ►A. Exercises
Graph.
7. x  2 and x  1 3 6 -3 -6

35. ►A. Exercises
Graph.
9. x  2 and x  1 3 6 -3 -6

36. ►A. Exercises
Graph.
11. x  2 and x  1 3 6 -3 -6

37. ►B. Exercises
Solve each inequality.
13. x + 6  5 and 3x + 4  7

38. ►B. Exercises
Solve each inequality.
15. |x| = 3

39. ►B. Exercises
Solve each inequality.
17. |x|  3

40. ►B. Exercises
Solve each inequality.
19. |x|  -3

41. ►B. Exercises
21. Consider the five inequalities; ,
, , , . Which inequalities
have the reflexive property?

42. ►B. Exercises
23. Consider the five inequalities; ,
, , , . Which inequalities
have the transitive property?

43. ►C. Exercises
25. Explain why 5  2.

44. ►C. Exercises
26. Explain why -7  -10.

45. ►C. Exercises
27. Show that 5  16 and then
conclude the 5  4.

46. ►C. Exercises
28. Show that 3  11

47. ■ Cumulative Review State each postulate or theorem.
31. Line Separation Postulate

48. ■ Cumulative Review State each postulate or theorem.
32. Theorem on perimeter of a regular n-gon.

49. ■ Cumulative Review State each postulate or theorem.
33. Ruler Postulate

50. ■ Cumulative Review State each postulate or theorem.
34. Midpoint Theorem

51. ■ Cumulative Review State each postulate or theorem.
35. Jordan Curve Theorem