I'd like to state that when a variable is between say $1$ and $2$, certain properties hold. Since this is a physical variable, the limits aren't precise. Is it acceptable to write: $$2 \gtrsim x \gtrsim 1$$ Although this seems like that intuitive symbol to use, I haven't seen it used anywhere.
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Sure, that's a fine symbol to use, as long as you explain what you intend it to mean.
For example, you could write
You could also use error bounds, e.g. $2\pm 0.1$, if you have such information regarding the imprecision you want to specify.