(1) If the solution to a linear programming problem exists, it will occur at a corner point. Theorem (Fundamental Theorem of Linear Programming). Notice the rolling pin must leave the feasible region at one of its corner points, in this case (0, 0) for a minimum value and (0, 72) for a maximum value. Think of these iso-m lines as a rolling pin. All of these lines are parallel with smaller M s giving lines closer to the origin and larger M s lines further from the origin. We now look to the objective function M = x + y and graph it on the feasible region for the cases where M = 20, 30, 40, 50, and 60. LINEAR PROGRAMMING Thus the corners of the feasible region are (0, 0), (50, 0), (0, 72), and (30, 32). For a true statement, the solution region is the side of the line that the test point is on for a false statement, it is the other side. Substitute this point into the linear inequality. To determine which side for an inequality, choose a test point not on the line (such as (0, 0) if the line does not pass through the origin). The graph of one inequality is all the points on one side of the line, the graph of the other all the points on the other side of the line. We are familiar with the graph of the linear equation.

Algebra of Inequalities Suppose x + 3 11 (each side is divided by -2 or multiplied by 4y and 24 there are 3 possibilities: 3x 4y = 24 3x 4y 24 4y = 3x y 3x + 24 y = 3 4 x 6 y > 3 4 x 6 y < 3 4 x 6 The three solution sets above are disjoint (do not intersect or overlap), and their graphs fill up the plane. Our focus will be on the nonstrict inequalities. The last two inequalities are called strict inequalities. x y means x is greater than or equal to y. Graphing Linear Inequalities x apple y means x is less than or equal to y.

Inequalities zoom math 500