1. Graphing Linear Inequalities in Two Variables Chapter 4 – Section 1

2. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 2 Expressions of the type x + 2y ≤ 8 and 3x – y > 6 are called linear inequalities in two variables. A solution of a linear inequality in two variables is an ordered pair (x, y) which makes the inequality true. Example: (1, 3) is a solution to x + 2y ≤ 8 since (1) + 2(3) = 7 ≤ 8. Solution of Linear Inequalities

3. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 3 The solution set, or feasible set, of a linear inequality in two variables is the set of all solutions. The solution set is a half-plane. It consists of the line x + 2y ≤ 8 and all the points below and to its left. The line is called the boundary line of the half-plane. Feasible Set Example: The solution set for x + 2y ≤ 8 is the shaded region. x y 2 2

4. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 4 x y x y If the inequality is, the boundary line is dotted; its points are not solutions. If the inequality is ≤ or ≥, the boundary line is solid; its points are solutions. Example: The boundary line of the solution set of x + y < 2 is dotted. Example: The boundary line of the solution set of 3x – y ≥ 2 is solid. 3x – y < 2 Boundary lines 3x – y = 2 3x – y > 2

5. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 5 x y Example: For 2x – 3y ≤ 18 graph the boundary line. The solution set is a half-plane. A test point can be selected to determine which side of the half-plane to shade. Shade above and to the left of the line. Use (0, 0) as a test point. (0, 0) is a solution. So all points on the (0, 0) side of the boundary line are also solutions. (0, 0) 2 -2 Test Point

6. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 6 To graph the solution set for a linear inequality: 2. Select a test point, not on the boundary line, and determine if it is a solution. 3. Shade a half-plane. 1. Graph the boundary line. Graphing an Inequality

7. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 7 x y Example: Graph the solution set for x – y > 2. 1. Graph the boundary line x – y = 2 as a dotted line. 2. Select a test point not on the line, say (0, 0). (0) – 0 = 0 > 2 is false. 3. Since this is a not a solution, shade in the half-plane not containing (0, 0). (0, 0) (2, 0) (0, -2) Example: Graph an Inequality

8. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 8 Solution sets for inequalities with only one variable can be graphed in the same way. Example: Graph the solution set for x < - 2. x y 4 4 - 4- 4 - 4- 4 x y 4 4 - 4- 4 - 4- 4 Example: Graph the solution set for x ≥ 4. Inequalities in One Variable

9. Linear Programming Chapter 4 – Section 2

10. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 10 Linear programming is a strategy for finding the optimum value – either maximum or minimum - of a linear function that is subject to certain constraints. Example: Find the maximum value of z, given: Definition: Linear Programming These constraints, or restrictions, are stated as a system of linear inequalities. z = 3x + 2y and Example continued objective function constraints

11. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 11 The system of linear inequalities determines a set of feasible solutions. The graph of this set is the feasible region. Example continued: Graph the feasible region determined by the system of constraints. Feasible Region x y y = 0 x = 0 feasible region Example continued

12. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 12 x y If a linear programming problem has a solution, then the solution is at a vertex of the feasible region. Example continued: Maximize the value of z = 2x +3y over the feasible region. Example: Feasible Region feasible region (0, 0) (0, 5) (7, 0) (5, 3) z = 2(0) + 3(0) = 0 z = 2(0) + 3(5) = 15 z = 2(7) + 3(0) = 14 z = 2(5) + 3(3) = 19 Test the value of z at each of the vertices. The maximum value of z is 19. This occurs at the point (5, 3) or when x =5 and y = 3. Maximum value of z

13. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 13 Solving Linear Programming Problems Graphically 1. Graph the feasible region. 2. Find the vertices of the region. 3. Evaluate the objective function at each vertex. 4. Select the vertices that optimize the objective function. a) If the feasible region is bounded the objective function will have both a maximum and a minimum. b) If the feasible region is unbounded and the objective function has an optimal value, the optimal value will occur at a vertex of the feasible region. Note: If the optimum value occurs at two vertices, its value is the same at both vertices and along the line segment joining them. Definition: Solving Linear Programming Problems Graphically

14. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 14 x y Example: Find the maximum and minimum value of z = x + 3y subject to the constraints –x + 3y  6, x –3y  6, and x + y  6. 1. Graph the feasible region. 2. Find the vertices. 3. Evaluate the objective function at each vertex. 4. The maximum value of z is 12 and occurs at (3, 3). The minimum value of z is 6 and occurs at both (0, 2) and (6, 0) and at every point along the line joining them. Example: Graphic Solution (0, 2) (3, 3) (6, 0) z = 0 + 3(2) = 6 z = 6 + 3(0) = 6 z = 3 + 3(3) = 12 Maximum value of z

15. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 15 Rewriting the objective function z = x + 3y in slope-intercept form gives. This equation represents a family of parallel lines, one for each value of z. As z increases through values 0, 3, 6, 9, 12, and 15, the corresponding line passes through the feasible region. The point at which the family of lines first meets the feasible region gives the minimum value of z, and the point at which the family of lines leaves the feasible region gives the maximum. Example: Using Parallel Lines y x (0, 2) (3, 3) (6, 0)

16. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 16 y x Example: Minimize z = 3x – y subject to x – y  1, x + y  5, x  0, and y  0. vertexvalue of z at vertex (0, 0)z = 3(0) – (0) = 0 (1, 0)z = 3(1) – (0) = 3 (3, 2)z = 3(3) – (2) = 7 (0, 5)z = 3(0) – (5) = –5 The minimum value of z is –5 and this occurs at (0, 5). Example: Minimum Value x = 0 y = 0 (0, 0) (1, 0) (3, 2) (0, 5)

17. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 17 x y Example: Maximize z = 2x + y subject to 3x + y  6, x + y  4, x  0, and y  0. Since the feasible region is unbounded there may be no maximum value of z. For x  4, (x, 0) is a feasible solution. At (x, 0), z = 2x. Therefore as x increases without bound, z increases without bound and there is no maximum value of z. Example: Maximum Value (0, 6) (1, 3) (4, 0)

18. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 18 Example: Sarah makes bracelets and necklaces to sell at a craft store. Each bracelet makes a profit of $7, takes 1 hour to assemble, and costs $2 for materials. Each necklace makes a profit of $12, takes 2 hour to assemble, and costs $3 for materials. Sarah has 48 hours available to assemble bracelets and necklaces. If she has $78 available to pay for materials, how many bracelets and necklaces should she make to maximize her profit? To formulate this as a linear programming problem: 1. Identify the variables. 2. Write the objective function. 3. Write the constraints. Example: Linear Programming problem Example continued

19. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 19 2. Express the profit as a function of x and y. p = 7x + 12y 3. Express the constraints as inequalities. Since Sarah cannot make a negative number of bracelets or necklaces, x  0 and y  0 must also hold. Time limitation: x + 2y  48. Example continued Example continued: Example continued Cost of materials: 2x + 3y  78. Function to be maximized 1. Let x = the number of bracelets Sarah makes Let y = the number of necklaces Sarah makes

20. Mathematical Applications by Harshbarger (8th ed) Copyright © by Houghton Mifflin Company 20 Example continued: Maximize p = 7x + 12y subject to the constraints 2x + 3y  78, x + 2y  48, x  0, and y  0. Sarah should make 12 bracelets and 18 necklaces for a maximum profit of $300. Example continued y x (0, 24) (12, 18) (39, 0) (0, 0) z = 0 z = 288 z = 273 z = 300