The document discusses rational inequalities and absolute value inequalities. It explains how to solve rational inequalities by simplifying the expression, factoring any quadratics, placing critical numbers on a number line, testing points in each interval, and stating the solution intervals. Examples are provided of solving rational inequalities algebraically and graphically. Absolute value inequalities are introduced and it is explained how the graphs of absolute value functions can be used to solve absolute value inequalities both graphically and algebraically. Practice exercises involving various types of inequalities are presented.

1. Inequality
1

2. Rational Inequality
x ­ 1
Example < 0
(x ­ 2)(x + 3)
1. Simplify the rational expression so that zero is on one side
and the expression involving x is on the other side
2. Factor any quadratic expressions
3. Place critical numbers on a number line
Critical Numbers: the zeros, and the values that make
the inequality undefined(non­permissible values)
4. Test points within each interval between the critical values, to
determine if the expression as a whole is positive or negative
5. State the intervals that qualify as solutions to the
inequality
2

3. Solve the inequality
x ­ 1
< 0
(x ­ 2)(x + 3)
­10 ­9 ­8 ­7 ­6 ­5 ­4 ­3 ­2 ­1 0 1 2 3 4 5 6 7 8 9 10
3

4. x ­ 1
The graph of y = (x ­ 2)(x + 3)
< 0
4

5. Solve:
x2 ­ 2x ­ 8
> 0
_
x ­ 1
­10 ­9 ­8 ­7 ­6 ­5 ­4 ­3 ­2 ­1 0 1 2 3 4 5 6 7 8 9 10
5

6. Graph of y = x2 ­ 2x ­ 8
x ­ 1
6

7. Solve
x 1
>
x ­ 3 x + 2
­10 ­9 ­8 ­7 ­6 ­5 ­4 ­3 ­2 ­1 0 1 2 3 4 5 6 7 8 9 10
7

8. 8

9. Absolute Value Inequality
Graph y = x Graph y = x
9

10. Graph y = x + 5 What will graph of
y = x + 5 look like?
10

11. 11

12. 12

13. 13

14. Solve graphically x ­ 2 <
_ 5
14

15. Solve algebraically x ­ 2 <
_ 5
15

16. x
+ 1 _
< 3
2
16

17. x
Solve algebraically + 1 _
< 3
2
17

18. Exercise 28
questions 6 ­ 12
18