1. Chapter 2: Equations and Inequalities
Section 2.2: Solving Inequalities

2. Section 2.2: Solving Inequalities
Goal: Students will be able to solve inequalities and graph the solution sets

3. Section 2.2: Solving Inequalities
Trichotomy Property – For any two real numbers a and b, exactly one of the following statements in true:
a < b
a = b
a > b

4. Section 2.2: Solving Inequalities
Transitive Property
For any real numbers a, b and c:
If a < b and b < c, then a < c.
Example:
If 2 < 5 and 5 < 9,
Then 2 < 9

5. Section 2.2: Solving Inequalities
Addition Property of Inequality
For any real numbers a, b and c:
If a > b, then a + c > b + c
If a < b, then a + c < b + c
Example
3 < 5
3 + (-4) < 5 + (-4)
-1 < 1

6. Section 2.2: Solving Inequalities
Subtraction Property of Inequality
For any real numbers a, b and c:
If a > b, then a – c > b – c
If a < b, then a – c < b – c
Example
2 > -7
2 – 8 > -7 – 8
-6 > -15

7. Section 2.2: Solving Inequalities
Example 1
Solve 4y – 3 < 5y Graph the solution set on a number line.

8. Section 2.2: Solving Inequalities
REMEMBER:
When multiplying or dividing each side by a NEGATIVE number, you must reverse the inequality symbol

9. Section 2.2: Solving Inequalities
Multiplication Property of Inequality
For any real numbers a, b and c where c is positive:
If a > b, then ac > bc
If a < b, then ac < bc
Example
-2 < 3
4(-2)< 4(3)
-8 < 12

10. Section 2.2: Solving Inequalities
Division Property of Inequality
For any real numbers a, b and c where c is positive:
If a > b, then a/c > b/c
If a < b, then a/c < b/c
Example
-18 < -9
-18/3< -9/3
-6 < -3

11. Section 2.2: Solving Inequalities
Division Property of Inequality
For any real numbers a, b and c where c is negative:
If a > b, then a/c < b/c
If a < b, then a/c > b/c
Example
12 > 8
12/-2 > 8/-2
-6 < -4

12. Section 2.2: Solving Inequalities
The solution set of an inequality can be expressed by using set-builder notation
{x | x > 9}
The set {} of all numbers x such that (|) x is greater than (>) 9

13. Section 2.2: Solving Inequalities
Example 3
Solve Graph the solution set on a number line.

14. Section 2.2: Solving Inequalities
Solve and graph 14 – 2x ≤ 4 (6 – x). Write the solution using set builder notation.

15. Section 2.2: Solving Inequalities
Solve and graph 3 (2x – 3) ≥ 5 (x – 3). Write the solution using set builder notation.

16. Section 2.2: Solving Inequalities
Homework:
Practice Exercises Pg. 55 #2-32 (even), #37, 38