Introduction to Feasible Regions

Feasible regions are defined as the set of all possible solutions to a system of linear inequalities that satisfy all the constraints in a linear programming problem.
Geometrically, a feasible region can be visualized on a graph as the common area shared by all inequalities.
Unbounded feasible regions extend indefinitely, while bounded feasible regions are confined within a certain region.

Characteristics of Feasible Regions

The vertices or corner points of the feasible region often contain the optimal solution for a linear programming problem.
Feasible regions can either be empty (when no solutions satisfy all constraints) or non-empty (when there exist solutions that satisfy all constraints).
The solution to a linear programming problem can be unique or there can be multiple optimal solutions.

To draw feasible regions, start by graphing each linear inequality on the same graph. Each inequality divides the graph into two halves.
Identify which half of the graph satisfies each inequality and find the overlap of all these areas. This overlap represents the feasible region.
Label the vertices of the feasible region for further analysis.

Linear programming problems often require finding the maximum or minimum value of an objective function within the feasible region.
Plot the objective function on the same graph as the feasible region. The optimal solution is usually one of the vertices of the feasible region.
If the objective function line passes through the feasible region, then there may be multiple optimal solutions. If it only touches the feasible region at a single vertex, then there is a unique optimal solution.
Solving linear programming problems often involves iterative methods, such as the Simplex method or graphical analysis.