In this paper we provide a relaxation result for control systems under both equality and inequality constraints involving the state and the control. In particular, we show that the Mangasarian-Fromowitz constraint qualification allows us to rewrite constrained systems as differential inclusions with locally Lipschitz right-hand side. Then the Filippov-Wazewski relaxation theorem may be applied to show that ordinary solutions are dense in the set of relaxed solutions. If, besides agreeing with the above constraints, the state has to remain in a control-independent set K, then the Mangasarian-Fromowitz condition cannot hold. This case is investigated as well by means of a condition on the feasible velocities on the boundary of K.