Interior-point methods for linear programming problems require the repeated solution of a linear system of equations. Solving these linear systems is non-trivial due to the severe ill-conditioning of the matrices towards convergence. This issue can be alleviated by incorporating suitable regularization terms in the linear programming problem. Regularization also allows us to efficiently handle rank deficient constraint matrices. We provide a convergence analysis of a regularized inexact interior-point method. The term `inexact' refers to the fact that we do not need to compute the true solution of the linear system of equations, only an approximation thereof. The formulation of the algorithm is sufficiently general such that specialized linear algebra routines developed in other work on inexact interior-point methods can also be incorporated in our regularized framework. In this work, we exploit the inexactness by using a mixed-precision solver for the linear system of equations. More specifically, we perform a Cholesky factorization in IEEE single precision and use it as a preconditioner for the Conjugate Gradient method. Numerical experiments illustrate the benefits of this approach applied to linear programming problems with a dense constraint matrix.