6.8 Linear Inequalities in Two Variables and Systems of Linear Inequalities 373 y x 5 6 4 3 2 1 1 2 3 4 5 6 7 (1, 7) (1, 1) (6, 7) 7 8 (6, 6) Figure 6.39 P x y 20 30 . = + Fig. 6.39 shows the feasible region. The feasible region consists of the shaded region and the boundaries. The vertices of the feasible region are at (1, 1), (1, 7), (6, 7), and (6, 6). Next we calculate the value of the objective function, P, at each one of the vertices. = + = + = = + = = + = = + = P x y P P P P At (1, 1), At (1, 7), At (6, 7), At (6, 6), 20 30 20(1) 30(1) 50 20(1) 30(7) 230 20(6) 30(7) 330 20(6) 30(6) 300 The maximum profit is at (6, 7). Therefore, the company should manufacture 6 washers and 7 dryers to maximize its profit. The maximum profit is $330. 7 Now try Exercise 53 Use the following steps to solve a linear programming problem. SOLVING A LINEAR PROGRAMMING PROBLEM 1. Determine all necessary constraints. 2. Determine the objective function. 3. Graph the constraints and determine the feasible region. 4. Determine the vertices of the feasible region. 5. Determine the value of the objective function at each vertex. The solution is the maximum or minimum value obtained when the coordinates of the vertices are substituted into the objective function. PROCEDURE Instructor Resources for Section 6.8 in MyLab Math • Objective-Level Videos 6.8 • Animation: Graphing Linear Inequalities with 2 Variables • PowerPoint Lecture Slides 6.8 • MyLab Exercises and Assignments 6.8 Exercises Warm Up Exercises In Exercises 1–10, fill in the blank with an appropriate word, phrase, or symbol(s). 1. The set of all points in a plane on one side of a line is called a(n) ________. Half-plane 2. When graphing an inequality in two variables, if the inequality contains either < or > use a(n) ________ line. Dashed 3. When graphing an inequality in two variables, if the inequality contains either or ≤ ≥ use a(n) ________ line. Solid 4. The set of points that satisfy all inequalities in a system of linear inequalities is called the ________ set. Solution 5. The solution set to a system of linear inequalities is the set of all ________ pairs that are common to all the inequalities in the system. Ordered 6. In a linear programming problem the restrictions that are represented by linear inequalities are called the ________. Constraints 7. In a linear programming problem, the shaded region formed by graphing the system of inequalities is called the ________ region. Feasible 8. The points of intersection of the boundaries of the feasible region are called ________. Vertices 9. In a linear programming problem, the function used for the quantity that we want to maximize or minimize is called the ________ function. Objective 10. The maximum and minimum values of the objective function occur at the ________ of the feasible region. Vertices Practice the Skills 11. Determine whether (0, 0) is a solution to the given inequality. a) x y 3 4 12 − > No b) x y 2 3 6 + ≤ Yes c) y x 2 3 5 ≥ − Yes d) y x 5 4 3 < − − No 12. Determine whether (0, 1) is a solution to the given inequality. a) x y 0 + ≥ Yes b) x y 3 2 0 − < Yes c) y x 2 3 < No d) y x 5 7 ≤ − No SECTION 6.8
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