In this paper, we propose an extension of the classical weight reduction inequalities for the binary knapsack polytope for settings where the maximum-weight item in the associated pack is not unique. We derive sufficient conditions under which the extended inequalities are facet-defining and identify conditions under which they strictly dominate the original weight reduction inequalities. In addition, we introduce a new class of valid inequalities for the binary knapsack polytope, named weight division inequalities. For the special class of binary knapsack set in which all items weighing less than half the knapsack capacity have the same weight, we show that its convex hull is completely characterized by weight reduction inequalities and weight division inequalities, along with the trivial nonnegativity constraints.