Let X and Y be the numbers of gallons of liquid C and D, respectively.


Then the objective function is  F(X,Y) = X + Y.


The constraint inequalities are

1*x + 6*Y <= 36     (1)  (red dye constraint)
2*X + 1*Y <= 28     (2)  (blue dye constraint)

X >= 0,  Y >= 0.


The feasibility region is shown in the figure below.

It is the quadrilateral in QI, adjacent to x- and y- axes and constrained by the red and the green lines.



  


Plots 1X + 6Y = 36 (red)  and  2X + Y = 28 (green)


The feasibility quadrilateral has the corner  

    P1 = (0,6)      (red line y-intercept);

    P2 = (12,4)     (intersection point of the red and green lines);  and

    P3 = (14,0)     (green line x-intercept).


The objective function has the values

     at P1:  F(X,Y) = 0 + 6 = 6;

     at P2:  F(X,Y) = 12 + 4 = 16;   and

     at P3:  F(X,y) = 14 + 0 = 14.


According to the Linear programming method, it means that the point P2 gives the solution to the given linear minimax problem:


    The most amount of gallons you can create of both color C and color D is 16.

    It happens when 12 gallons of the liquid C and 4 gallons of the liquid D is produced.