This study is concerned with the development and implementation of a preconditioner for a set of hyperbolic partial differential equations resulting from a new 5-moment closure for the prediction of gas flows both in and out of local equilibrium. This new 5-moment closure offers a robust and efficient system of first-order hyperbolic partial differential equations that has proven to provide an accurate treatment of one-dimensional gases, both in and for significant departures from local thermodynamic equilibrium. However, numerical computations using this model have proven to be difficult as a result of a singularity in the closing flux of the system. This also causes infinitely large wavespeeds in the system. The main goal of this work is to mitigate these numerical issues. Since the solution of a hyperbolic system is characterized by the waves of the system, one could suggest to scale these wavespeeds to remove the arbitrarily large speeds without altering the solution of the system. To accomplish this, this work starts with a detailed study of the behaviour of the system’s wavespeeds, given by the eigenvalues of the flux Jacobian of the system. Since, it is not possible to solve for these eigenvalues explicitly, it is suggested to approximate them by interpolation between the few states at which these waves can be solved for explicitly. With an estimate for the wavespeeds, the nature of the singularity in the system can be analyzed mathematically. The results of this mathematical analysis are used to develop a preconditioner matrix to remove the singularity from the model. To implement the proposed preconditioned model numerically, a centred-difference scheme with artificial dissipation is proposed. A dual-time-stepping strategy is developed and implemented with implicit Euler time marching for both physical and pseudo time iteration. This dual-time treatment allows the preconditioned system to remain applicable to time-accurate problems and is found to greatly increase the robustness of the solution of the steady-state problems. Solutions to several canonical problems for both continuum and non-equilibrium flow are computed and comparisons are made to classical models.