How about
FullSimplify[(1/2 (7 - 3 Sqrt[5]) < x <=
2 - Sqrt[2] && -(1/(-7 x + x^2)) < y < 1) || (2 - Sqrt[2] < x <
1 && ((-2 + x)/(2 (-7 + x)) -
1/2 Sqrt[(-4 + 8 x - 4 x^3 + x^4)/((-7 + x)^2 x^2)] <
y < (-2 + x)/(2 (-7 + x)) +
1/2 Sqrt[(-4 + 8 x - 4 x^3 +
x^4)/((-7 + x)^2 x^2)] || -(1/(-7 x + x^2)) < y < 1)),
Assumptions -> 0 < x < 1 && 0 < y < 1]
1 + (-7 + x) x y < 0 || (-2 + x + Sqrt[-4 + 8 x - 4 x^3 + x^4]/x)/( 2 (-7 + x)) < y < (-2 + x - Sqrt[-4 + 8 x - 4 x^3 + x^4]/x)/( 2 (-7 + x))
? Addition.
RegionPlot[1 + (-7 + x) x y < 0 || (-2 + x + Sqrt[-4 + 8 x - 4 x^3 + x^4]/x)/
(2 (-7 + x)) < y < (-2 + x - Sqrt[-4 + 8 x - 4 x^3 + x^4]/x)/(2 (-7 + x)),
{x, 0, 1}, {y, 0, 1}, PlotPoints -> 50]
RegionPlot[1 + (-7 + x) x y < 0, {x, 0, 1}, {y, 0, 1},PlotPoints -> 50]
RegionPlot[(-2 + x + Sqrt[-4 + 8 x - 4 x^3 + x^4]/x)/(2 (-7 + x)) < y < (-2 + x - Sqrt[-4 + 8 x - 4 x^3 + x^4]/x)/(2 (-7 + x)), {x, 0, 1}, {y, 0, 1}, PlotPoints -> 50]
