1. Exercise
Solve for x, telling what property was used to solve the equation.
x − 3 = 7
x = 10; Addition Property of Equality

2. Exercise
Solve for x, telling what property was used to solve the equation.
x + 3 = 7
x = 4; Addition Property of Equality

3. Exercise
Solve for x, telling what property was used to solve the equation.
3x = 12
x = 4; Multiplication Property of Equality

4. Exercise
Solve for x, telling what property was used to solve the equation.
= 12
x 3
x = 36; Multiplication Property of Equality

5. Exercise
Solve for x, telling what property was used to solve the equation.
−x = 7
x = −7; Multiplication Property of Equality

6. Example 1
Graph x > 6. −2 2 4 6 8 10

7. Example 2
Graph x ≤ −3. −6 −4 −2 2 4 6

8. Example 3
Graph x ≠ 2. −6 −4 −2 2 4 6

9. Addition Property of Inequality
If a and b are real numbers, such that a < b, and c is any real number, then a + c < b + c.

10. Example 4 Solve x − 5 ≥ −2, and graph the solution set. x − 5 ≥ −2
x ≥ 3 −2 2 4 6 8 10

11. Example 5 Solve x + 8 < −12, and graph the solution set.
− 8 − 8
x < −20
−30
−20
−10 10

12. Example 6
Solve 14 > x + 9.
14 > x + 9
− − 9
5 > x
x < 5

13. Multiplication Property of Inequality
If a and b are real numbers, such that a < b, and c is any positive real number, then ac < bc.

14. Multiplication Property of Inequality
If a and b are real numbers, such that a < b, and c is any negative real number, then ac > bc.

15. Reverse the inequality sign only when multiplying or dividing by a negative.

16. Example 7
Solve −5x ≥ 35.

17. Example 7
y 3
Solve < −4.

18. Example 8 Solve 2x − 9 < 15. 2x − 9 < 15 + 9 + 9 2x < 24
2x < 24
x < 12

19. Example 9 x 4 Solve − + 7 ≥ 20. x 4 − + 7 ≥ 20 − 7 − 7 x 4 − ≥ 13 x 4
− ≥ 20
x 4
− 7 − 7
− ≥ 13
x 4
−4( ) (13)−4
− ≥
x 4
x ≤ −52

20. Example
Graph x = 3. −4 −2 2 4

21. Example
Graph x ≥ 6. 2 4 6 8

22. Example
Graph x ≠ −2. −4 −2 2 4

23. Example
Graph x < 5. −2 2 4 6

24. Example
Solve.
8x ≥ 16
x ≥ 2

25. Example
Solve.
15 ≤ −5x
−5 −5
−3 ≥ x

26. Example
Solve.
−6( ) ≥
x −6
(−10)−6
x −6
≥ −10
x ≤ 60

27. Example
Solve.
9x < 81
x < 9

28. Example
x 2
Solve 20 ≤
20 ≤
x 2
− − 8
12 ≤
x 2 2
24 ≤ x

29. Example
Solve.
−4x − 5 > 15
x < −5

30. Exercise What integers would make the inequality x < 5 true?
any integers whose distance from the origin is less than 5: −4, −3, −2, −1, 0, 1, 2, 3, 4

31. Exercise What values of x would make the inequality x > 3 true?
any number whose distance from the origin is more than 3: that is, any number x such that x > 3 or x < −3