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This is my model:

http://www.everymac.com/systems/apple/imac/specs/imac-core-2-duo-2.4-20-inch-aluminum-specs.html


If I install 4gb of RAM myself, can I upgrade the OS to Mountain Lion? Thanks!


Sure as long as it meets or exceeds the minimum specifications for ML it will be fine. You have installed 4GB, you could upgrade to 6GB which would make the experience even better. As long as you are currently running Snow Leopard 10.6.6 or later upgrading is as simple as downloading and following the on-screen instructions. As with ANY major upgrade ensure you have a good backup, for example I would recommend both a Time Machine and bootable clone backup on separate external HDs.
 
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