Advanced Mathematical Concepts Chapter 2 Lesson 2-6 Example 1 CANDY Jennifer is making a party mix using two different kinds of candy, candy corn and licorice. She wants the total amount of candy in the mix to be no more than five pounds and wants to use at least 3 pounds of licorice. What combinations of amounts of candy corn and licorice will satisfy Jennifer’s requirements? First, write two inequalities that represent each of Jennifer’s requirements. Let c represent the number of pounds of candy corn and l represent the number of pounds of licorice. Total: c+l5 Licorice: l3 Both of these inequalities include the boundary line, so the lines are solid. The graph of c + l 5 is composed of all points on and below the line c + l = 5. The graph of l 3 includes all points on and above the line l = 3. The orange area is the solution to the system of inequalities. That is, the ordered pair for any point in the orange area satisfies both inequalities. Example 2 a. Solve the system of inequalities by graphing. x2 y4 x–y3 b. Name the coordinates of the vertices of the polygonal convex set. a. Since each inequality contains an equality, the boundary lines will be solid. The shaded region shows points that satisfy all three inequalities. b. The region is a triangle whose vertices are the points (2, -1), (7, 4), and (2, 4). Advanced Mathematical Concepts Chapter 2 Example 3 Find the maximum and minimum values of f(x, y) = 2x + y for the polygonal convex set determined by the system of inequalities. x -2 x+y8 y1 x–y≤2 Boundary a Boundary b Boundary c Boundary d x -2 x+y 8 y1 x-y ≤2 y -x + 8 -y ≤ -x + 2 y ≥x-2 Graph the inequalities and find the coordinates of the vertices of the resulting polygon. The coordinates of the vertices are (-2, 1), (-2, 10), (5, 3), and (3, 1). Now, evaluate the function f(x, y) = 2x + y at each vertex. f(-2, 1) = 2(-2) + 1 or –3 f(-2, 10) = 2(-2) + 10 or 6 f(5, 3) = 2(5) + 3 or 13 f(3, 1) = 2(3) + 1 or 7 The maximum value of the function is 13, and the minimum value is –3.
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