There is no such thing as a cheap gaming computer. Try a used
computer, if it isn't too old, it should be pretty good for gaming.
Alienware is a good choice.
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For a gaming computer you will need at least 4GB of RAM. Most
people average 6GB for their gaming computers.
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A gaming computer is similar to a normal PC computer but is
often optimized for the gaming experience, for example a gaming
computer may have a higher quality of graphics, more memory, a
bigger hard drive etc.