[en] The discovery of a two-dimensional metal-insulator transition (MIT) has intensified the experimental and theoretical research of the two-dimensional electron gas. Recently the measurement of the compressibility has been considered as a detection method of the MIT. In this work the compressibility of the system is studied in the deformable jellium model (DJM) with the self-consistently Hartree-Fock (HF) approximation. In the DJM the background is statically deformed in order to get local charge neutrality. This method describes in a unified form, the properties of the system at the metallic, intermediate and low density regions, which includes the symmetry transition from the homogeneous phase in the metallic state to the localized phase in the insulating state. The single-particle state functions are taken as the usual plane wave multiplied by modulating functions with hexagonal symmetry. The coefficients of the expansion are determined self-consistently by solving the HF equations with orthonormality condition. At high densities the solution converges to a homogeneous system. After the transition point the solution corresponds to a localized system with a periodic density centered on a hexagonal lattice, which at low densities approaches to the Wigner crystal. A carefully calculation, with a great number of points, was made in the interest region in order to obtain a state equation for the system. The coupling parameter evaluated as the ratio of the average potential energy to the average kinetic energy, is used by the theoretical analysis as a precise criterion to determine the transition point, nevertheless the chemical potential is the property that can be utilized in the experimental praxis to locate precisely the critical point. Therefore the total energy (per unit area), the chemical potential and the compressibility are calculated for the densities range near to the transition from the fluid to the localized phase. Our results describe the qualitative behavior of the compressibility obtained for the two-dimensional hole system in GaAs/AlGaAs heterostructures in the metal-insulator transition. (Authors)