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If I do a region plot of the following inequality,

RegionPlot[(4 - y^2)^2 ((8 + y^2) - 18 x y + 9 x^2) > 
36 (1 - y^2) ((4 + y^2) - 8 x y + (4 + y^2) x^2) && x >= y, {x, 0, 
1}, {y, 0, 1}]

Mathematica plots a region where it is valid:

soln_exists.

However, if I change x >=y to x>y inside the RegionPlot command, the region vanishes altogether. I know that the diagonal line is x=y, but there clearly is a finite region where x>y is true and Mathematica does not displays that. Am I missing something?

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You need to increase the number of plot points RegionPlot uses:

inequalities = (4-y^2)^2 ((8+y^2)-18 x y+9 x^2)>36 (1-y^2) ((4+y^2)-8 x y+(4+y^2) x^2) && x> y;
RegionPlot[
    inequalities,
    {x,0,1},
    {y,0,1},
    PlotPoints->100
]

enter image description here

You could also use region functionality:

reg = ImplicitRegion[inequalities, {x, y}];
DiscretizeRegion[reg, {{0, 1}, {0, 1}}, Axes->True]

enter image description here

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  • $\begingroup$ Thank you! But why the option plot points matter for showing the existence of the region? this seems to be quite dangerous, to be honest! Or, is this a standard feature. Sorry, I am a newbie. $\endgroup$ – anotherone Jul 19 '19 at 15:09
  • $\begingroup$ Also, is there any other important matter that I should be aware of before plotting regions? Anyway to crosscheck? This time I got suspicious because I randomly checked x=0.9 and y=0.89 and found it to be consistent! $\endgroup$ – anotherone Jul 19 '19 at 15:11

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