Basic Question about Thomson and Compton Scattering

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Ok so I am doing a project on the derivation of the Compton Effect, and I would like to briefly discuss Thomson scattering, but I'm having a little trouble understanding what happens in Thomson scattering.

This is what I think I understand so far:
When a photon enters matter and interacts with electrons, the electron will be accelerated due to the magnetic and electric field of the photon. The electron will, in turn, emit radiation at the same frequency as the photon. So is this elastic scattering? And is this an interaction between the photon and ALL the electrons in the matter?

I'm also a little confused about Compton scattering. I've read that it is an example of inelastic scattering, but I know that in order to derive the Compton Effect I need to assume an elastic interaction between the photon and electron. Does inelastic scattering just mean that the frequency of the incident photon and the reradiated photon are different?
 

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