366 CHAPTER 6 Algebra, Graphs, and Functions A manufacturer of gas grills needs to determine how many grills to ship to each of two stores. The manufacturer can ship at most 300 grills in total. A linear inequality in two variables can be used to represent the number of grills that can be shipped to each store. In this section, we will discuss how to set up and solve linear inequalities in two variables. Linear Inequalities in Two Variables and Systems of Linear Inequalities SECTION 6.8 LEARNING GOALS Upon completion of this section, you will be able to: 7 Graph linear inequalities in two variables. 7 Graph systems of linear inequalities. 7 Solve linear programming problems. Why This Is Important Linear inequalities can be used to model problems similar to the one described above and many others that we encounter daily. Furthermore, inequalities are used in a branch of mathematics called linear programming, which can be used in maximizing a company’s revenue and profits or in minimizing the company’s costs. Linear programming is used in many occupations. Graphing Linear Inequalities in Two Variables In Section 6.5, we introduced linear inequalities in one variable. Now we will introduce linear inequalities in two variables. Some examples of linear inequalities in two variables are x y x y 2 3 7, 7 5, + ≤ + ≥ and x y3 6. − < The solution set of a linear inequality in one variable may be indicated on a number line. The solution set of a linear inequality in two variables is indicated on a coordinate plane. An inequality that is strictly less than ( )< or greater than ( )> will have as its solution set a half-plane. A half-plane is the set of all the points on one side of a line. An inequality that is less than or equal to ( )≤ or greater than or equal to ( )≥ will have as its solution set the set of points that consists of a half-plane and a line. To indicate that the line is part of the solution set, we draw a solid line. To indicate that the line is not part of the solution set, we draw a dashed line. Research Activity 73. The Rhind Papyrus The Rhind Papyrus indicates that the early Egyptians used linear equations. Do research and write a paper on the symbols used in linear equations and the use of the linear equations by the early Egyptians. 71. In parts (a)–(d), make up a system of linear equations whose solution will be the ordered pair given. (Hint: It may be helpful to visualize possible graphs that have the given solution. There are many possible answers for each part.) a) (0, 0) b) (1, 0) c) (0, 1) d) (1, 1) Answers will vary. 72. When solving a system of linear equations by the substitution method, a student obtained the equation 0 0 = and gave the solution as (0, 0). What is the student’s error? * *See Instructor Answer Appendix Nd3000/Shutterstock
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