The Kaczmarz method is an iterative algorithm that solves overdetermined systems of linear equalities. This paper studies a system of linear equalities and inequalities. We use the block version of the Kaczmarz method applied towards the equalities with the simple randomized Kaczmarz scheme for the inequalities. This primarily involves combining Needell and Tropp's work on the block Kaczmarz method with the application of a randomized Kaczmarz approach towards a system of equalities and inequalities performed by Leventhal and Lewis. We give an expected linear rate of convergence for this kind of system and find that using the block Kaczmarz scheme for the equalities can improve the rate compared to the simple Kaczmarz method.