When we run sensitivity analysis on the objective coefficient of XXX, the computer reports an allowable range of [9,20][9,20][9,20]. This means as long as the profit contribution of XXX stays within 920, the current optimal basis (the binding constraints and therefore the optimal X,YX^*,Y^*X,Y) does not change. If we raise the profit on XXX from 12 to 15(which lies in this range), the optimal decision variables remain exactly the same. Only the objective value changes by the marginal increase times the current optimal level of XXX:NewProfit=8,000+(1512)X=8,000+3X.\text{New Profit}\;=\; 8{,}000\;+\; (15-12)\,X^*\;=\; 8{,}000\;+\; 3X^*.NewProfit=8,000+(1512)X=8,000+3X.Because XX^*X is unchanged, this gives a straightforward dollar increase without re-solving the model.If we instead increase the profit on XXX to 25, we exceed the upper bound of 20. Crossing that bound invalidates the current basis, so the identity of the binding constraints can change and the optimal corner point must be recomputed. Intuitively, making XXX so profitable pushes the solution toward corners with more XXX (subject to feasibility), potentially reducing YYY. The net effect is that both the optimal variables and the total profit can change in a non-linear (piecewise) way relative to the coefficient change; you must re-optimize to find the new X,YX^{**}, Y^{**}X,Y and profit.On every LP with a feasible region has an infinite number of solutions: with continuous decision variables, any polyhedral feasible region (a line segment, polygon, or polytope) contains uncountably many feasible points, so there are infinitely many feasible solutions. However, the optimal solution need not be infinite: it is typically a single extreme point (unique optimum), a whole edge/face (multiple optima when the objective is parallel to a binding constraint), or unbounded (no finite optimum).