Feasible Region 
The feasible region determined by a collection of linear inequalities is the collection of points that satisfy all of the inequalities.
To sketch the feasible region determined by a collection of linear inequalities in two variables: Sketch the regions represented by each inequality on the same graph, remembering to shade the parts of the plane that you do not want. What is unshaded when you are done is the feasible region. 
Example 
The feasible region for the following collection of inequalities is the unshaded region shown below (including its boundary). 
3x - 4y ≤ 12, 

    x + 2y ≥ 4 

    x ≥ 1 

    y ≥ 0.
    

        

            

            

            
Graphical Method 
            

                
The graphical method for solving linear programming problems in two unknowns is as follows. 
                
A. Graph the feasible region. 
                
B. Compute the coordinates of the corner points. 
                
C. Substitute the coordinates of the corner points into the objective function to see which gives the optimal value. 
                
D. If the feasible region is not bounded, this method can be misleading: optimal solutions always exist when the feasible region is bounded, but may or may not exist when the feasible region is unbounded. The textbook shows a straightforward way for determining whether optimal solutions exist in the case of unbounded feasible regions.
                
If you want to see a utility that automates the whole process, try our Linear Programming Grapher.
                
It does everything automatically!