Summary: We consider a very simple integer program involving production of a single item and start-up costs for the standard machines first studied by Lasdon and Terjung. Solving directly as an integer program leads to prohibitively large branch and bound trees. Here, we show how using a simple family of valid inequalities and a heuristic procedure to choose one of these inequalities as a cut, it is possible to reduce substantially the size of the tree, and in several cases to eliminate the need for branch and bound. The valid inequalities used are all simple Gomory cuts. However, they are derived from the initial problem formulation rather than from the optimal linear programming tableau.