Example #1:   y  ≥  –x – 1
                       y  ≤  2x + 1

The intersection of both graphs occurs in the darkened area because of the shading.

Shading occurs above the red line and to the right and below the blue line. 

The solution to the system of Inequalities,  y  ≥  –x – 1 and  
y  ≤  2x + 1, is the intersection
of both graphs.

Example #2:  y  ≥  –x – 1                 
                      y  ≤  2x + 1
                      x < 1

Shading occurs above the red line, to the right and below of the blue line and to the left of the green line.

1.)  find the equations for the boundary lines:

           Determine the slope , and then use the point-slope form of  y – y₁ =  m(x – x₁) 
           to find the equation of the line.

2.)  make sure each boundary line has the appropriate inequality symbol

          ≤,  ≥ will be a solid line
          <,  > will be a dashed line

Example #3:  The blue line represents x  ≥  0 and the red line represents
y  ≥  0.  To find the equations of
the green and pink lines, find the slope, and then use y – y₁ =  m(x – x₁) to get the equation.

     The green line contains the points       (0, 1) and (2, 3) so the slope is:

     Choose one of the given points, (2, 3), to find the equation.

           y – 3 = 1(x – 2)
           y – 3 = x – 2
           y = x + 1

     Since the green line is dashed and shaded under the line, the equation is  y < x + 1.

     The pink line contains the points (3, 0) and (1, 3) so the slope is

           

     Choose one of the points, (3, 0), to find the equation.

           

     Since the pink line is solid and shaded below the equation is .

Linear programming is a method used to find a minimum or maximum value of some quantity.  Often this quantity is cost or profit and used in business applications. 

Constraints are the inequalities used in linear programming.

Feasible region is the solution set.

Objective function is the function to be maximized or minimized.

Example #1:  Graph the following constraints.  Name the coordinates of the vertices of the feasible region.  Find the maximum and minimum values for the objective function  P = 3x + y.

          Constraints:
                               x ≥ 1
                               y ≥ 0
                               2x + y ≤ 6

          Step 1:   Graph the inequalities.

                     x ≥ 1                                              y ≥ 0                                     2x + y ≤ 6
                                                                                                                                y ≤ –2x + 6


When the inequalities are all graphed in the same coordinate plane, we can determine the feasible region (solution set of points), the region that is overlapped by all three graphs.   

The intersection of the boundaries are the vertices of the feasible region. 

Step 2:  Find the coordinates of the vertices of the feasible region. 

The intersection of the boundaries are the vertices of the feasible region.  The polygon formed is a triangle with vertices at (1, 4), (3, 0) and (1 ,0).

Step 3:  Evaluate the objective function at each vertex.

          P = 3x + y

          At (1, 4):       P = 3(1) + 4 = 7

          At (3, 0):       P = 3(3) + 4 = 9 (Maximum)

          At (1, 0):       P = 3(1) + 4 = 3  (Minimum)

P has a maximum value of  9 at (3, 0) and a minimum value of 3 at (1, 0).

Example #2:  A crafter is making two sizes of wreath arrangements.  The crafter charges $14 for the small arrangement and $20 for the large.  She wants to make at least 15 small and 10 large arrangements with a total of no more than 50 arrangements.  Let x represent the number of small arrangements, and let y represent the number of large arrangements.  How many of each should she make to maximize her revenue?

The question asks for the optimal combination of wreath arrangements to maximize the crafter’s revenue.
          x:  number of small arrangements
          y:  number of large arrangements

Constraints:

          x ≥ 15           - She wants to make at least ( ≥ ) 15 small arrangements.

          y ≥ 10           - She wants to make at least ( ≥ ) 10 large arrangements.

          x  + y ≤ 50    - She wants to make at total of no more than ( ≤ ) 50 large arrangements.

Solve x  + y ≤ 50 for y.

          x + y ≤ 50
        –x           –x  
          y ≤ –x + 50

Here is the graph displayed in the coordinate plane.  Only the area that is overlapping between all three inequalities is shaded.

*The scale of the graph is 10 units per line. *Notice the shading occurs to the right of the red, above the blue and below the green lines.  This will yield 3 vertices at (15, 10), (15, 35) and (40, 10).

The objective function that will represent the maximum profit is:

          P = 14x + 20y because each small wreath costs $14 and each large wreath costs $20.

Replace each of the vertices in for x and y to find the maximum value.

Since the maximum value is 910, the crafter should make 15 small wreaths and 35 large wreaths to gain maximum revenue.