Since the simplex method requires the polyhedron to be in the positive domain, the 1-norm minimization problems are formulated by substantially increasing the size of the LP problems. This paper presents a simple modification that enables the simplex method to be directly applicable to a polyhedron, which extends into the negative domain. That is, instead of requiring the problem to change, the method is changed to I fit the problem. The modification eliminates the need to increase the size of the problem and thus eliminates the associated computational I effort. The proposed method skips iterations and Phase 1 of the simplex method. Its computational advantage is verified in two examples.