Any way to solve parametric inequations?

09142021, 10:58 PM
(This post was last modified: 09152021 02:30 AM by jte.)
Post: #13




RE: Any way to solve parametric inequations?
As a human, trying to understand xy/(x+y)^2≤0, for real x and y, one approach would be to handle (0,0) separately first and then look at the punctured plane, R^2\{(0,0)}.
… so… doing that… Realizing that (x,y) is a solution iff (kx,ky) is, for positive k, lets us change our focus from R^2\{(0,0)} to the unit circle (each point on the circle is then a representative for an open ray from the origin), or — better yet — for nonzero k, which lets us consider half of a unit circle (each point on the half circle is then a representative for a punctured line through the origin). If we pick the northeast half circle (x^2+y^2=1 s.t. y≥x … although we only need one of the half circle’s end points), we can then exploit the symmetry that (x,y) is a solution iff (y,x) is to fold the half circle over y=x and consider just the upper quarter of a unit circle, x^2+y^2=1 s.t. y≥.5sqrt(2) (now each point is a representative of two punctured lines, or we could unfold our solution later before expanding back to the punctured plane). Looking at xy/(x+y)^2, one can see where the solutions are (from the left endpoint, not including it, to the halfway point, including it). I’m not quite sure how this relates to automated solving / automated presentation of results. But considering symmetries etc. is certainly part of mathematics. 

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