1. 6.6
Linear Inequalities

2. Symbols of Inequality a < b means that a is less than b.
a  b means that a is less than or equal to b.
a > b means that a is greater than b.
a  b means that a is greater than or equal to b.
Find the solution to an inequality by adding, subtracting, multiplying or dividing both sides by the same number or expression.
Change the direction of the inequality symbol when multiplying or dividing both sides of an inequality by a negative number.

3. Example: Graphing
Graph the solution set of x  4, where x is a real number, on the number line.
The numbers less than or equal to 4 are all the points on the number line to the left of 4 and 4 itself. The closed circle at 4 shows that 4 is included in the solution set.

4. Example: Graphing
Graph the solution set of x > 3, where x is a real number, on the number line.
The numbers greater than 3 are all the points on the number line to the right of 3. The open circle at 3 is used to indicate that 3 is not included in the solution set.

5. Practice Problem : Graph on a number line
5 > x
7 – x < 3

6. Example: Solve and graph the solution
Solve 3x – 8 < 10 and graph the solution set.

7. Practice Problem: Solve and graph on a number line
2x – 5 > 7
12 > 3x +15

8. Compound Inequality
Graph the solution set of the inequality 4 < x  3
a) where x is an integer.
The solution set is the integers between 4 and 3, including 3.

9. Compound Inequality continued
Graph the solution set of the inequality 4 < x  3
b) where x is a real number
The solution set consists of all real numbers between 4 and 3, including the 3 but not the 4.

10. Practice Problem: solve and graph
-3 < x < 4

11. Example
A student must have an average (the mean) on five tests that is greater than or equal to 85% but less than 92% to receive a final grade of B. Jamal’s grade on the first four tests were 98%, 89%, 88%, and 93%. What range of grades on the fifth test will give him a B in the course?

12. Example continued

13. Practice problem

14. Homework
p. 333 # 9 – 57 odd