Solving a set of equations/inequations

[Han]

I am of the opinion that inequality "solvers" are generally quite useless. Here is what Maple provides as the solution to:

solve({x^2+y^2>3, x^2+y^2<4});

{-2 < y, x < sqrt(-y^2+4), y < -sqrt(3), -sqrt(-y^2+4) < x}, {y = -sqrt(3), -1 < x, x < 0}, {y = -sqrt(3), 0 < x, x < 1}, {x < -sqrt(-y^2+3), y < sqrt(3), -sqrt(3) < y, -sqrt(-y^2+4) < x}, {x < sqrt(-y^2+4), y < sqrt(3), -sqrt(3) < y, sqrt(-y^2+3) < x}, {y = sqrt(3), -1 < x, x < 0}, {y = sqrt(3), 0 < x, x < 1}, {x < sqrt(-y^2+4), y < 2, -sqrt(-y^2+4) < x, sqrt(3) < y}

This, while correct, provides absolutely no insight into the solution that a simple graph would convey. Your specific example is more of an exception rather than a representation of the typical system of inequalities one could encounter.

EDIT: For the record, Wolfram Alpha gives an equally useless solution. The original inequalities themselves are much more meaningful than the "solutions" once one recognizes that they (the original inequalities) represent regions inside/outside a circle.

solve({3*x+y>4,2*x-y<1});

{ 1 < y, x < (1/2)*y+1/2, -(1/3)*y+4/3 < x }

Again, correct yet mostly useless. This answer requires knowledge about lines written in the atypical form of \( x=My+B\) as opposed to \(y=mx+b\) so that one can geometrically realize the solution. But if you look at it, the solution is not really that much more useful than the original inequalities themselves. In fact, the best that most solvers would be able to do in a typical system of inequalities is to solve for one variable (say \( y \)) in all the formulas and then simply insert "and" in between each formula. It is really no different than the solution above (in the second example).

If you want meaningful solutions to a set of inequalities, I would recommend using the Advanced Graphing app. A picture is worth a thousand words (or formulas).