1. Section 6.6 Linear Inequalities

2. What You Will Learn Solving Linear Inequalities
Solving Compound Inequalities

3. 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.

4. Inequality
An inequality consists of two (or more) expressions joined by an inequality sign.

5. Solving Inequalities
To indicate the solution set of x < 2 on the number line, we draw an open circle at 2 and a line to the left of 2 with an arrow at its end.
The open circle indicates that the solution set does not include the number 2.

6. Solving Inequalities
To indicate the solution set of x ≤ 2 on the number line, we draw a closed circle at 2 and a line to the left of 2 with an arrow at its end.
The closed circle indicates that the solution set does not include the number 2.

7. Solving Inequalities
Find the solution to an inequality by adding, subtracting, multiplying or dividing both sides by the same number or expression.
Reverse the direction of the inequality symbol when multiplying or dividing both sides of an inequality by a negative number.

8. Example 4: Dividing by a Negative Number
Solve the inequality –5x < 20 and graph the solution set on the number line.
Solution

9. Compound Inequality
An inequality of the form a < x < b is called a compound inequality.
It means that a < x and x < b.

10. Example 7: A Compound Inequality
Graph the solution set of the inequality –3 < x ≤ 2 where x is an integer.
Solution
The solution set is the integers between –3 and 2, including 2.

11. Example 7: A Compound Inequality
Graph the solution set of the inequality –3 < x ≤ 2 where x is a real number.
Solution
The solution set is all real numbers between –3 and 2, including 2.

12. Example 9: Average Grade
A student must have an average (the mean) on five tests that is greater than or equal to 80% but less than 90% to receive a final grade of B. Devon’s grades on the first four tests were 98%, 76%, 86%, and 92%. What range of scores on the fifth test will give him a B in the course?

13. Example 9: Average Grade
Solution
Let x = the fifth grade

14. Example 9: Average Grade
Solution
A grade of 48% up to but not including a grade of 98% on the fifth test will result in a grade of B in this course.