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In Linear Programming, we use the “origin side” method to determine which side is the “>” side of the straight line $f(x, y) = 0$.

I know it is NOT necessary true that $f(x, y) > 0$ is always on the “right hand side” of $f(x, y) = 0$. But it seems if we re-write $f(x, y) = 0$ with $((x)) > 0$, that guess is true.

Question:- Is it the case? If it is a yes, then is it a standard method same as the “origin side” and how can we convince others?

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The cases of identifying the regions separated by the lines (y = constant or x = constant) are too trivial to consider. Hence, we can assume the slanted line is $L : y = mx + c$ with $m \ne 0$.

Re-write the equation as $L : x = \dfrac {1}{m}y + d$; where $d = \dfrac {c}{m}$.

Let D be another constant such that $D \gt d$. Then, the point $(D, 0)$ is NOT on $L$ and in fact it is on the right hand side of $L$.

After putting $x = D$ and $y = 0$ in $L$, we can say that the points lying on the same side as $(D, 0)$ (i.e. the right hand side of $L$) is represented by $x \gt \dfrac {1}{m}y + d$.

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