Systems of Inequalities. Linear Programming What you’ll learn To solve systems of linear inequalities. To solve real world problems involving systems of linear inequalities. To solve problems using linear programming. Vocabulary Constrain, linear programming, feasible region, objective function.

Take a note An inequality and a system of inequalities can each have many solutions. A solution of a system of inequalities is a solution for each inequality of the system. You can solve a system of inequalities in more then one way. Graphing the solution is usually the most appropriate method. The solution is the set of all points that are solutions of each inequality in the system. When solving a system of linear inequalities by graphing. A graphed solution consist of a half a plane and possibly a boundary line ,so the solution is the overlap of the two half planes.

This is the solution for the system Solving a system by graphing. What is the solution of the system of inequalities? This is the solution for the system

What is the solution of the system of inequalities? Your turn What is the solution of the system of inequalities? Solution of the system y ≥x-3 y-3x≤-4 y ≥x-3 Answer: y-3x≤-4

Your turn again Solving a linear/absolute value system What is the solution of the system of inequalities? x y

and the inside is called feasible region Linear programming Constrains (borders) and the inside is called feasible region Graph X=0 then y=10 (0,10) y=0 then x=10 (10,0) What is the solution for all inequalities at once?

If there is a maximum or a minimum value of the linear Take a note The constrains in a linear programming Situation form a system of inequalities, like the one you did before. The graph of the system (the purple part) is the feasible region. It contains all the points that satisfy all the constrains. The quantity you are trying to maximize or minimize is modeled with an objective function. Often this quantity is cost or profit. If there is a maximum or a minimum value of the linear objective function, it occurs at one or more vertices of the feasible region. Some real world problems involved multiple linear relationships. Linear programming accounts for all these linear relationships and give the solution to the problem.

Testing vertices. Multiple choice Linear programming Testing vertices. Multiple choice What point in the feasible region maximizes P for the objective function P=2x+y Answer: A.(2,0) B.(0,0) C.(3,1) D.(0,2.5) Step 1: graph the inequalities. Step 2: form the feasible region Step 3: find the coordinates of each vertex Step 4: evaluate P at each vertex.

x=0 then y=-2 and y=0 then x=2 Coordinates of each vertex First Q. A(2,0) B(0,0) C(3,1) D(0,2.5) Evaluate P=2(0)+0=0 P=2(0)+2.5=2.5 P=2(3)+1=7 P=2(2)+0=4 C Maximum value D B A Answer: C

Classwork odd Homework even