1. Warm-up:
Solve x2 – 3x + 2 = 0
HW: pg (2, 6, 8, 10, 12, 28, 32, 42, 44, 50, 66, 70, 74, 76)

2. P.6 Solving Inequalities

3. Objective:
Solve and graph linear inequalities.
Solve and graph absolute value inequalities.
Solve and graph polynomial inequalities.
Write solutions in interval notation.

4. Greater than or equal to
Inequality Symbols
Less than
Not equal to
Less than or equal to
Greater than or equal to
Greater than

5. Ex: Solve the inequality.
Flip the sign if multiplying or dividing by a negative number!

6. Graphing Linear Inequalities
Remember:
< and > signs will have ), ( or an open dot o
and signs will have ], [ or a closed dot
graph of graph of ) [ 4 5 6 7 -3 -2 -1
(-, 11/2) Unbounded
[-2, ) Unbounded

7. Example: Solve and graph the solution.] 6 7 8 9
(-, 7] Unbounded

8. Compound Inequality Example: Solve & graph. -18 ≤ 2t+8 ≤ 20[ ]
-13 6
[-13, 6] Bounded

9. Solve & graph. -6x + 9 < 3 or -3x -8 ≤ 13 -6x < -6 -3x ≤ 21[ ( [ -7 1
[-7,) Unbounded

10. Absolute Value Inequalities
Recall:

11. Absolute Value Inequalities
Case 1 Example:
Less th”and”
and ( ) ( ) -2 8
(-2, 8) Bounded

12. Absolute Value Inequalities
Case 2 Example:
Great “or” or OR ] [ -5 4
(-, -5] U [4, ) Unbounded

13. Polynomial Inequalities

14. Definition of a Polynomial Inequality
Polynomial Inequalities
Definition of a Polynomial Inequality
A polynomial inequality is any inequality that can be put in one of the forms
where f is a polynomial.

15. Procedure for Solving Polynomial Inequalities
1) Express the inequality in the form f (x) < 0 or f (x) > 0
2) Solve the equation f (x) = 0. The real solutions are the boundary points.
3) Locate these boundary points on a number line, dividing the number line into intervals.
4) Choose one representative (test) number within each interval and check to see if it satisfies the inequality.
5) Write the solution set, selecting the interval(s) that satisfy the given inequality.

16. Polynomial Inequalities
EXAMPLE
Solve and graph the solution set on a number line:
SOLUTION
1) Express the inequality in the form f (x) < 0.
2) Solve the equation f (x) = 0.
The x-intercepts of f are 3 and 1. We will use these x-intercepts as boundary points on a number line.

17. Polynomial Inequalities
3) Locate the boundary points on a number line and separate the line into intervals. The number line with the boundary points is shown as follows:
The boundary points divide the number line into three intervals:

18. Polynomial Inequalities
4) Choose one representative (test) number within each interval and check to see if it satisfies the inequality.
Interval
Test Number
Check
Conclusion

19. Polynomial Inequalities
CONTINUED
5) Write the solution set, selecting the interval(s) that satisfy the given inequality.
Thus, the solution set of the given inequality, , is
The graph of the solution set on a number line is shown as follows: ( )

20. Polynomial Inequalities
EXAMPLE
Solve and graph the solution set on a number line:
SOLUTION
1) Express the inequality in the form f (x) 0.

21. Polynomial Inequalities
2) Solve the equation

22. Polynomial Inequalities
3) Locate the boundary points on a number line and separate the line into intervals. The number line with the boundary points is show as follows:
The boundary points divide the number line into three intervals:

23. Polynomial Inequalities
4) Choose one representative (test) number within each interval and check to see if it satisfies the inequality.
Interval
Test Number
Check
Conclusion

24. Polynomial Inequalities
5) Write the solution set
Careful! Because the inequality involves , we must include the solutions of namely 2 and -2, in the solution set.
Thus, the solution set of the given inequality, is
The graph of the solution set on a number line is shown as follows: [

25. Summary:
Solve and graph linear inequalities.
Solve and graph absolute value inequalities.
Solve and graph polynomial inequalities.
Write solutions in interval notation.

26. Sneedlegrit: Solve x2 – 6 < x
HW: pg (2, 6, 8, 10, 12, 28, 32, 42, 44, 50, 66, 70, 74, 76)