Graphing Systems of Inequalities Lesson 7-5 Graphing Systems of Inequalities

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Objectives Solve systems of inequalities by graphing Solve real-world problems involving systems of inequalities

Vocabulary System of inequalities - a set of two or more inequalities with the same variables

Solving Systems of Inequalities Solve each inequality separately and graph The final solution set is always the intersection of the two (or more) solution spaces (as shown below in purple) x + y > 4 ½x – y > -2 y ≥ 0 x ≥ 0 y x First Quadrant Only!

Example 1 Solve the system of inequalities by graphing. Answer: The solution includes the ordered pairs in the intersection of the graphs of and The region is shaded in green. The graphs and are boundaries of this region. The graph is dashed and is not included in the graph of . The graph of is included in the graph of

Example 2 Solve the system of inequalities by graphing. Answer: The graphs of and are parallel lines. Because the two regions have no points in common, the system of inequalities has no solution.

Example 3 Service A college service organization requires that its members maintain at least a 3.0 grade point average, and volunteer at least 10 hours a week. Graph these requirements. Words The grade point average is at least 3.0. The number of volunteer hours is at least 10 hours. Variables If the grade point average and the number of volunteer hours, the following inequalities represent the requirements of the service organization.

Example 3 cont Inequalities The grade point average is at least 3.0. The number of volunteer hours is at least 10. Answer: The solution is the set of all ordered pairs whose graphs are in the intersection of the graphs of these inequalities.

Example 4 Employment Jamil mows grass after school but his job only pays $3 an hour. He has been offered another job as a library assistant for $6 per hour. Because of school, his parents allow him to work 15 hours per week. How many hours can Jamil mow grass and work in the library and still make at least $60 per week? Let the number of hours spent mowing grass and the number of hours spent working in the library. Since g and both represent a number of days, neither can be a negative number. The following system of inequalities can be used to represent the conditions of this problem.

Example 4 cont The solution is the set of all ordered pairs whose graphs are in the intersection of the graphs of these inequalities. Only the portion of the region in the first quadrant is used since and . Answer: Any point in the region is a possible solution. For example (2, 10) is a point in the region. Jamil could mow grass for 2 hours and work in the library for 10 hours during the week.

Summary & Homework Summary: Homework: Graph each inequality on a coordinate plane to determine the intersection of the graphs Homework: Pg 397 12-28 even