Feasible Regions Explained with Simple Animations

Now we'll see how multiple inequalities work together to create a bounded region.Let's start with our first inequality: y is less than or equal to three x plus two.Next, we add a second inequality: y is greater than or equal to negative x plus one.The region that satisfies both inequalities is their intersection, shown here in purple.Notice how each point in this region must satisfy both inequalities simultaneously.Let's add a third constraint: x is less than or equal to three.This new constraint further restricts our feasible region, shown here in yellow.Every point in our final yellow region must satisfy all three inequalities at the same time.These boundary points are particularly important, as they often represent optimal solutions in real-world applications.Now that we have our feasible region, let's see how to determine which points represent valid solutions to our furniture factory problem.Our factory makes chairs and tables. The x-axis represents chairs, and the y-axis represents tables.The shaded region represents all possible combinations of chairs and tables we can make within our constraints.Let's test some points. The point (2, 3) means making 2 chairs and 3 tables.This point satisfies all our constraints: it's within our daily limits and uses less than 18 labor hours.Similarly, the point (4, 2) is also valid, representing 4 chairs and 2 tables.However, the point (5, 4) is not valid. While it's within our chair limit, it exceeds our labor hour constraint.The point (7, 1) is also invalid because it exceeds our maximum chair production of 6 chairs per day.Let's verify why (5, 4) is invalid by checking each constraint.Any point within the shaded region, like (3, 3), represents a valid production plan that satisfies all our constraints.