MAT 141 ‐ Chapter 3 Finite Mathematics MAT 141: Chapter 3 Notes Graphing Linear Inequalities Linear Programming David J. Gisch Linear Inequalities Graphing with Intercepts • Find the � -intercept. ▫ Substitute � � 0 and solve for � . • Find the y-intercept. ▫ Substitute x � 0 and solve for � . Example: Graph the equation using the intercepts. 2� � 4� � 12 1
MAT 141 ‐ Chapter 3 Graphing with Intercepts Graphing with Intercepts Example: Graph the equation using the intercepts. Example: Graph the equation using the intercepts. 150� � 300� � 15,000 2� � 3� � 12 Inequalities Inequalities Graph the equation using the intercepts. Graph the equation using the intercepts. 2� � 3� � 12 2� � 3� � 12 2
MAT 141 ‐ Chapter 3 Which way do you S hade S olid Line or Dotted Line? 2 Methods. • If there is an equals sign then it is a solid line. 1. Test a point. ▫ Substitute a point (not on the line) into the • If there is not an equals sign then it is a dotted line. equation. If it results in a true inequality shade that side, if not, shade the other side. 2. Solve for Y and follow the sign. �, � means you shade up ▫ ▫ �, � means you shade down 3� � 12� � 12 Graphing with Intercepts S ystems of Linear Inequalities Example: Graph the inequality. • You graph all of the inequalities and where all of the shaded regions overlap is the solution. �6� � 15� � 30 3
MAT 141 ‐ Chapter 3 S ystems of Linear Inequalities Linear Inequalities Example: Graph the system of inequalities. � � � � 1 Graph � 2� � 3� � 12 � � �2 � � 1 � 2� � � � 8 Linear Inequalities Linear Inequalities Example: Graph the system of inequalities. Example: Graph the system of inequalities. � � 0 � � 0 � � 0 � � 0 � � 800� � 2000� � 400 2� � 5� � 10 � � 2� � 8 4
MAT 141 ‐ Chapter 3 Feasible Region Example: The Gillette company produces two popular electric razors, the M3Power™ and the Fusion Power ™. Due to demand, the number of M3Power razors is never more than half the number of Fusion Power razors. The factories production cannot exceed more than 800 razors per day. (a) Write a system of inequalities to express the conditions of the Gillette Company. (b) Graph the feasible region. Feasible Region • As briefly mentioned in the previous section, the graph of a system of linear inequalities is called referred to as a feasible region. ▫ All the points in that region are scenarios that meet the limitations of our constraints. Solving Linear Programming Problems Graphically 5
MAT 141 ‐ Chapter 3 Linear Programming Equations with Multiple Variables • If an equation has two variables it is an equation that can • Linear Program m ing is a method for solving be graphed in the �� plane, hence 2D. problems in which a particular quantity that must be ▫ If the variables have no powers then it is a linear equation. maximized or minimized is limited by other factors, ▫ Example, � � 5� � 10 or � � � 5� � 10 called constraints . • If an equation has three variables it is an equation that can • An objective function is an algebraic expression in be graphed in space, hence 3D. two or more variables describing a quantity that must be ▫ If the variables have no powers then it is a plane (flat surface). maximized or minimized. ▫ Example, z � 3� � 12� or � �, � � 3� � 12� Obj ective Function Constraints Example: Bottled water and medical supplies are to be Example: Each plane can carry no more than 80,000 shipped to victims of an earthquake by plane. Each pounds. The bottled water weighs 20 pounds per container of bottled water will serve 10 people and each container and each medical kit weighs 10 pounds. Let x medical kit will aid 6 people. Let x represent the number represent the number of bottles of water to be shipped of bottles of water to be shipped and y the number of and y the number of medical kits. medical kits. Write the objective function that describes the number of people that can be helped. (a) Write an inequality that describes this constraint. (b) Are there any other constraints? 6
MAT 141 ‐ Chapter 3 Feasible Region Obj ective Functions and Feasible Regions • The feasible region is the inputs, that fit our constraints, Example: Graph the feasible region from example 2. for the objective function. • The lowest point (minimum) or highest point (maximum) of the graph of the objective function will be at a corner. Feasible Region Check the Points Example: What are the corner points of the region in Corner Points Objective Function example 2? � �, � � ��� � �� 0, 0 10 0 � 6 0 � 0 0, 8000 10 0 � 6 8000 � 48,000 4000, 0 10 4000 � 6 0 � 40,000 20� � 10� � 80,000 While it seems odd, we can help the most people if we ship zero containers of water and 8,000 medical kits. Of course if we wanted to include some minimum amount of water that would add another constraint and change our feasible region. 7
MAT 141 ‐ Chapter 3 S teps for Linear Programming Applications of Linear Programming Applications of Linear Programming Did Y ou Know? • In the last section we had few constraints and therefore the corner points were easy to find. • We now will look at having several constraints and using systems of equations to find the corner points. 8
MAT 141 ‐ Chapter 3 Flower Arranging Flower Arranging Example: Flowers Unlimited has two spring floral (b) Write down the constraints. 10� � 5� � 120 arrangements, the Easter Bouquet and the Spring 20� � 20� � 300 Bouquet. The Easter Bouquet requires 10 jonquils and 20 � daisies and produces a profit of $1.50. The Spring � � 0 Bouquet requires 5 jonquils and 20 daisies and yields a � � 0 profit of $1. How many of each type of arrangements should the florist make to maximize the profit if 120 (c) Graph the feasible region. jonquils and 300 daisies are available? (a) Write an objective function. � �, � � 1.5� � 1� Flower Arranging Flower Arranging (d) Find the corner points. (e) Substitute the corner points into the objective function. 9
MAT 141 ‐ Chapter 3 Flower Arranging Flower Arranging Example: A construction company needs to hire at least (b) Write down the constraints. � � � � 100 100 employees for a project. They will need at least 30 � � � � 30 more unskilled laborers than skilled laborers. At least 20 skilled laborers should be hired. The unskilled laborers � � 20 earn $8 per hour, and the skilled laborer earns $15 per � � 0 hour. How many employees should the company hire to � � 0 minimize its hourly cost while satisfying all of the requirements? (c) Graph the feasible region. (a) Write an objective functions. � �, � � 8� � 15� Flower Arranging Flower Arranging (d) Find the corner points. (e) Substitute the corner points into the objective function. 10
MAT 141 ‐ Chapter 3 Flower Arranging Flower Arranging Example: Certain animals at a rescue shelter must have at (b) Write down the constraints. least 30 g of protein and at least 20 g of fat per feeding 2� � 6� � 30 ����� �� ������� period. These nutrients come from food A, which cost 18 4� � 2� � 20 ����� �� ��� � cents per unit and supplies 2 g of protein and 4 g of fat; � � 2 ���� � �������� and food B, which cost 12 cents per unit and supplies 6 g of protein and 2 g of fat. Food B is bought under a long (c) Graph the feasible region. term contract requiring at least 2 units of B be used per serving. How much of each type of food must be bought to minimize the cost per serving. (a) Write an objective functions. � �, � � .18� � .12� Flower Arranging Flower Arranging (d) Find the corner points. (e) Substitute the corner points into the objective function. 11
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