1.6 Solving Inequalities.

1. 1.6Solving Inequalities.

2. Trichotomy Property a<ba=ba>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. [ ] a b

10. An open interval, denoted (a, b), consists of all real numbers x for which a<x<b. ( ) a 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. [ ) 0 2 -3

17. 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.

18. Inequalities Involving Absolute Value

19. Solve absolute value inequality Solution set:(-4/3, 2) ( ) 0 3 -2 1 -1 2

20. Inequalities Involving Absolute Value

21. Solve Inequality