Section 6.6 Linear Inequalities

What You Will Learn Solving Linear Inequalities Solving Compound Inequalities

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.

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

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.

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.

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.

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

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

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.

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.

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?

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

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.