The inelastic x-ray scattering by L and M electrons in the vicinity of the argon K-absorption edge is analyzed with the use of a generalization of the Kramers-Heisenberg formula. The theoretical cross section is given as a superposition of scattering probabilities to final (mp)-1np, n>=4 discrete and (mp)-1ɛp continuum states for m=2 and 3. The effect of the beamwidth on the spectrum of the scattered photons is discussed in detail. For a highly monochromatized beam the resolution is shown to increase by a factor of Γ1sΓmp as compared to K photoabsorption measurements. Our work demonstrates the possibility of using the inelastic resonance scattering for a detailed study of Rydberg and exciton states in the x-ray region.