SOLUTION: I would like to know if it is correct that the domain of x-4/2x-3 would be (-infinity, 4) U (4, infinity)

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Not quite correct. You want to find the values where the denominator end up being non-zero, not the numerator.
Domain is all x except when (2x-3)=0
x = 3/2 makes the denominator. So x cannot be 3/2
(-infinity, 3/2) U (3/2, infinity)