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3.2 Graphical Solutions                                                                         73


                   3.2     Graphical Solutions


                   Before giving a more general algorithm for handling this problem and prob-
                   lems like it, we note that when the number of variables is small (preferably 2),
                   a graphical technique can be used.
                      Inequalities, such as the four given in Pablo’s problem, are often called
                   constraints, and values of the variables that satisfy these constraints comprise
                   the so-called feasible region. Since there are only two variables, this is easy
                   to plot:


                   Example 36 (Constraints and feasible region) Pablo’s constraints are

                                                       x ≥ 5
                                                       y ≥ 7

                                                  15 ≤ x + y ≤ 25 .
                   Plotted in the (x, y) plane, this gives:
































                      You might be able to see the solution to Pablo’s problem already. Oranges
                   are very sugary, so they should be kept low, thus y = 7. Also, the less fruit
                   the better, so the answer had better lie on the line x + y = 15. Hence,
                   the answer must be at the vertex (8, 7). Actually this is a general feature


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