Author 
Comment/Response 
Bill Simpson

01/20/13 8:31pm
Turn each of your inequalities into an equality.
Thus x+y<7 becomes x+y==7.
For every pair of equalities use Solve to find the unique intersection.
Thus {x,y}/.Solve[{3x+y==7, 2xy==2}, {x,y}][[1]]
The Outer[] function might help you do this.
http://reference.wolfram.com/mathematica/ref/Outer.html?q=Outer&lang=en
Use ConvexHull on the resulting set of points.
http://reference.wolfram.com/mathematica/ComputationalGeometry/ref/ConvexHull.html
You might try doing this one small step at a time, while verifying that each step gives a correct result, before trying everything at once.
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