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I don't think I can do a better job than the OG, but here goes:
(1) you know that sqrt(4x) is an integer. You can split sqrt(4x) into two factors:
sqrt(4x) = sqrt(4)*sqrt(x)
We know that sqrt(4)=2 (integer), so the only way for sqrt(4)*sqrt(x) to be an integer is for sqrt(x) to be an integer. Otherwise sqrt(4x) would be a non-integer, and we would have a contrdiction with the statement.
(2) Similar reasoning to above...
sqrt(3x) = sqrt(3)*sqrt(x) is not an integer, which means that sqrt(x) is not an odd exponent of sqrt(3), which in turn means that x does not equal 3, 27, 243 and so on.
If we assume that x=4, statement (2) holds and sqrt(x) is an integer.
If we assume that x=5, statement (2) holds, but sqrt(x) is not an integer.