Section 2.8 Linear Inequalities 183 Instructor Note: Some students may be acquainted with the use of open and closed circles, ● or ●, used as endpoints on their graphs. The solution set to an inequality is the set of real numbers that make the inequality a true statement. For example, the solution set to the inequality is all real numbers 2 or greater. Because the solution set has an infinite number of values, we cannot list all of the individual solutions. However, we can graph the solution set on the number line. The square bracket symbol, , is used on the graph to indicate that the point is included in the solution set. By convention, square brackets, either or , are used to include a point on a number line. Parentheses, ( or ), are used to exclude a point on a number line. The solution set of the inequality includes the real numbers greater than 2 but not including 2. Therefore, a ( symbol is used on the graph to indicate that is not included. ) x 7 2 6 5 4 3 2 1 0 1 2 3 4 5 6 x 2 x 7 2 x 2 x 2 6 5 4 3 2 1 0 1 2 3 4 5 6 x 2 In Example 1, we demonstrate how to graph linear inequalities. To graph an inequality means that we graph its solution set. That is, we graph all of the values on the number line that make the inequality true. Graphing Linear Inequalities Example 1 Graph the solution sets. 7 3 a. b. c. x 7 1 Solution: a. 1. c 3 7 y ) 6 5 4 3 2 1 0 1 2 3 4 5 6 x 7 1 The solution set is the set of all real numbers strictly greater than Therefore, we graph the region on the number line to the right of 1. Because x 1 x 1. is not included in the solution set, we use the ( symbol at c . 73 b. is equivalent to 2 13 2 3 4 5 6 213 . 6 5 4 3 2 1 0 1 c 213 The solution set is the set of all real numbers less than or equal to Therefore, graph the region on the number line to the left of and including c 213 Use the symbol to indicate that is included in the solution set. 213 . c. This inequality reads “3 is greater than y.”This is equivalent to saying, “y is less than 3.” The inequality 3 y can also be written as y 3. ( 6 5 4 3 2 1 0 1 2 3 4 5 6 3 7 y y 6 3 The solution set is the set of real numbers less than 3. Therefore, graph the region on the number line to the left of 3. Use the symbol ) to denote that the endpoint, 3, is not included in the solution. Skill Practice Graph the solution sets. 1. y 6 0 5 2. x 4 3. 5 a Classroom Examples: p. 193, Exercises 6 and 10 Instructor Note: Students sometimes confuse the meaning of to think it means the set x 7 1 {2, 3, 4 . . . }. They don’t think about the real number line. They don’t see that 1.00000001 is a solution. Answers 1. 2. 3. 5 54 ( 0
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