1. 1.5 Solving Inequalities

2. Trichotomy Property
a < b a = b a > b

3. Nonnegative Property

4. Transitive Property of Inequalities
If a < b and b < c,then a < c.
If a > b and b > c, then a > c.

5. Addition Property of Inequalities
If a < b, then a + c < b + c.
If a > b, then a + c > b + c.

6. Multiplication Property of Inequalities
If a < b and if c > 0, then ac < bc.
If a < b and if c < 0, then ac > bc.
If a > b and if c > 0, then ac > bc.
If a > b and if c < 0, then ac < bc.

7. 3 < 5
2(3) < 2(5)
3 < 5
-2(3) > -2(5)
-6 > -10

8. Reciprocal Property for Inequalities

9. A closed interval denoted by [a, b], consists of all real numbers x for which a < x < b.

10. An open interval, denoted (a, b), consists of all real numbers x for which a < x < b.

11. A half-open, or half-closed interval is (a, b], consisting of all real numbers x for which a < x < b. ( ] a b

12. A half-open, or half-closed interval is [a, b), consisting of all real numbers x for which a < x < b. [ ) a b

13. [ a

14. ( a

16. Write the inequality -3 < x < 2 using
interval notation. Illustrate the inequality
using a real number line. [ ) 2 -3

17. Is equivalent to two inequalities:
Solve the inequality -3 < 2x - 7 < 5 and draw a graph to illustrate the solution.
-3 < 2x - 7 < 5
Is equivalent to two inequalities:
-3 < 2x and x - 7 < 5
< 2x
4 < 2x
2 < x
2x <5 + 7
2x <12
x < 6
and 2 6

18. Steps for Solving Inequalities Graphically
Write the inequality in one of the following forms:
Graph Y1 and Y2 on the same screen.
If inequality is of the form Y1 < Y2, determine on what interval Y1 is below Y2.
Similarly for Y1 > Y2.
If inequality is not strict include the endpoints of the intervals.

19. Inequalities Involving Absolute Value

20. Solve absolute value inequality
Solution set: (-4/3, 2) ( ) -2 -1 1 2 3

21. Inequalities Involving Absolute Value

22. Solve Inequality