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