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Kuba
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ContourPlot[x^3 - y^2 == 2 y, {x, -1, 1}, {y, -1, .1},
            RegionFunction -> (#^2 + #2^3 > 2 # &)]

enter image description here

The following procedure contains additional condition which you've provided by range for RegionPlot: {x, -1, 1}, {y, -1, .1}.

Reduce[x^3 - y^2 == 2 y && x^2 + y^3 > 2 x]
(x==-1&&y==-1)||
(-1<x<Root[-8-2x #1+5&& #1^2-3Abs[x] #1^3-#1^4+#1^5+#1^6&,1]&&
<= (y==-1-Sqrt[1+x^3]||y==-1+Sqrt[1+x^3]))||
(Root[-8-2 #1+5&& #1^2-31 #1^3-#1^4+#1^5+#1^6&<= y <= .1,1]<=x<0&&y==-1+Sqrt[1+x^3])|| 
(x>Root[-8-2 #1+5 #1^2-3 #1^3-#1^4+#1^5+#1^6&    {x,2]&&y==-1+Sqrt[1+x^3]) y}, Reals]
-1. <= x < 0 && y == -1. + Sqrt[1. + x^3]
ContourPlot[x^3 - y^2 == 2 y, {x, -1, 1}, {y, -1, .1},
            RegionFunction -> (#^2 + #2^3 > 2 # &)]

enter image description here

Reduce[x^3 - y^2 == 2 y && x^2 + y^3 > 2 x]
(x==-1&&y==-1)||
(-1<x<Root[-8-2 #1+5 #1^2-3 #1^3-#1^4+#1^5+#1^6&,1]&&
 (y==-1-Sqrt[1+x^3]||y==-1+Sqrt[1+x^3]))||
(Root[-8-2 #1+5 #1^2-3 #1^3-#1^4+#1^5+#1^6&,1]<=x<0&&y==-1+Sqrt[1+x^3])||
(x>Root[-8-2 #1+5 #1^2-3 #1^3-#1^4+#1^5+#1^6&,2]&&y==-1+Sqrt[1+x^3])
ContourPlot[x^3 - y^2 == 2 y, {x, -1, 1}, {y, -1, .1},
            RegionFunction -> (#^2 + #2^3 > 2 # &)]

enter image description here

The following procedure contains additional condition which you've provided by range for RegionPlot: {x, -1, 1}, {y, -1, .1}.

Reduce[x^3 - y^2 == 2 y && x^2 + y^3 > 2 x && Abs[x] <= 1 && -1 <= y <= .1, 
       {x, y}, Reals]
-1. <= x < 0 && y == -1. + Sqrt[1. + x^3]
Source Link
Kuba
  • 137.7k
  • 13
  • 289
  • 751

ContourPlot[x^3 - y^2 == 2 y, {x, -1, 1}, {y, -1, .1},
            RegionFunction -> (#^2 + #2^3 > 2 # &)]

enter image description here

Reduce[x^3 - y^2 == 2 y && x^2 + y^3 > 2 x]
(x==-1&&y==-1)||
(-1<x<Root[-8-2 #1+5 #1^2-3 #1^3-#1^4+#1^5+#1^6&,1]&&
 (y==-1-Sqrt[1+x^3]||y==-1+Sqrt[1+x^3]))||
(Root[-8-2 #1+5 #1^2-3 #1^3-#1^4+#1^5+#1^6&,1]<=x<0&&y==-1+Sqrt[1+x^3])||
(x>Root[-8-2 #1+5 #1^2-3 #1^3-#1^4+#1^5+#1^6&,2]&&y==-1+Sqrt[1+x^3])