But in order to assemble tables and chairs in the real world, both constraints must be met simultaneously. This new triangle denotes the feasible region with respect to the small-block constraint. Here the axes intercepts are (4, 0) and (0, 4), and the required half-plane is again to the lower left of the line. The procedure must be repeated to plot the other (small-block) constraint: 2x + 2y ≤ 8. We plotted the first structural constraint. (In fact, in order to consider the entire continuous area within a feasible region, we are compelled to work with the real numbers.) The discussion on Integer Programming will deal with the practical issues concerning integer solutions. Although one would be hard pressed to assemble a marketable fractional chair in the real world, we will accept fractional solutions in principle. As we shall see when we discuss the LP assumptions, in order for the algorithm to work properly, the decision variables must be allowed to assume non-integer values. Not all of the points in the triangle have integer coordinates. We see here how LP goes about in finding a solution: it first removes from consideration all points (production mixes) that are not feasible. Points outside the triangle are infeasible and thus not candidates for solution. The coordinates of any point within the triangle, edges included, represent combinations of tables and chairs ( x and y ) that can be produced given the amount of large blocks available. The light-blue triangle represents the region where production is feasible (possible) relative to the large-block availability constraint. (This half-plane is reduced to a triangle because the nonnegativity constraints apply.) Since the coordinate values satisfy the inequality, the lower-left triangle is the half-plane specified by the constraint. A good check point here is the origin (0, 0). If they do, then the point lies on the required half-plane if they don’t, then the other half-plane is the one specified. One can always determine which half-plane is specified by choosing a point not on the line and checking if its coordinates satisfy the inequality. Clearly, there are only two possibilities: the lower-left triangle and the upper-right unbounded region. We now select the half-plane specified by the original inequality: 2x + y ≤ 6.