A low-density expansion of the solution of the mean spherical approximation (MSA) for ion-dipole mixtures in the semi-restricted case (with equal size hard sphere cations and anions and different size hard sphere dipoles) is proposed. From the expansion, the explicit approximations of the three energy parameters (b(0), b(1), b(2)) and the thermodynamic properties are obtained. A reasonable explanation is offered as to why the dielectric constant of the vacuum cannot be used in the long-range term in the successful EOSs for real electrolytes solutions, although the dipolar solvent is treated as explicit molecule in the models. The energy of the solvation and the relationship between the primitive model (PM) and the nonprimitive model (NPM) are discussed in some limit cases. Comparisons of our approximation with the molecular simulation are also carried out. An EOS with rather simple expressions based on the expansion is applied to the calculations of activity coefficients, osmotic coefficients, and densities of the sodium chloride aqueous solutions.