The Fundamental Theorem of Linear Programming states that the maximum (or minimum) value of the objective function always takes place at the vertices of the feasible region.
The shaded region where all conditions are satisfied is the feasible region or the feasible polygon. Therefore, in this example, we shade the region that is below and to the left of both constraint lines, but also above the x axis and to the right of the y axis, in order to further satisfy the constraints \(x \geq 0\) and \(y \geq 0\). In the graph below, after the lines representing the constraints were graphed, the point (0,0) was used as a test point to determine that If the test point does not satisfy the inequality, then the region that satisfies the inequality lies on the opposite side of the line from the test point.If the test point satisfies the inequality, then the region of the plane that satisfies the inequality is the region that contains the test point.Any point on the plane that is not on the line can be used as a test point. A test point is used to determine which portion of the plane to shade to satisfy the inequality.
The line for a constraint will divide the plane into two region, one of which satisfies the inequality part of the constraint. However often the easiest method is to graph the line by plotting the x-intercept and y-intercept. In order to solve the problem, we graph the constraints and shade the region that satisfies all the inequality constraints.Īny appropriate method can be used to graph the lines for the constraints.
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