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