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]

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))

?

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]