I have made a rather obvious yet peculiar observation while calculating with quadratic inequalities. Take a simple quadratic inequality like the one below
by multiplying both sides by $x$, then subtracting $x$ from both sides we get
Hence, both the above inequalities are one and the same in theory. However, the solution to the first inequality is $x>0$, while the second inequality is satisfied for all $x\in\mathbb R$.
It is not difficult to explain this simple case, seeing that the LHS of the first inequality will turn negative for negative $x$, but my question is more of a general nature, if you like. I wonder, why do two identical (but paraphrased/rewritten) inequalities have different solutions?