Recently a new derivative-free algorithm  has been proposed for the solution of linearly constrained finite minimax problems. Such an algorithm can also be used to solve systems of nonlinear inequalities in the case when the derivatives are not available and provided that a suitable reformulation of the system into a minimax problem is carried out. In this paper we show an interesting property of the algorithm proposed in  when it is applied to the solution ofsystems of nonlinear inequalities. Namely, under some mild conditions regarding the regularity of the functions defining the system, it is possible to prove that the algorithm locate a solution of the problem after a finite number of iterations. Furthermore, under a weaker regularity condition, it is possible to show that an accumulation point of the sequence generated by the algorithm exists which is a solution of the system. The reported numerical results, though preliminary, confirm the good properties of the method.