[Please please please...post actual cut-and-pastable code.]

Here is a method that is, unfortunately, impractical. But it sometimes gives results if you are patient.

```
isEmpty[a_?NumericQ, b_?NumericQ] := Module[{finst},
finst =
FindInstance[(3*x + y Exp[x*y])*(x \[Minus] a) + (6*y +
x*Exp[x*y])*(y \[Minus] b) < 0, {x, y}];
If[ListQ[finst],
If[Length[finst] == 0, True, False]
, $Failed]
]
In[306]:= isEmpty[1, 3]
Out[306]= False
```

Here is a start on a method that uses contpur plotting. One must settle for a finite range on {x,y} for this; I use -+10 for both.

```
isEmpty2[a_?NumericQ, b_?NumericQ] := Module[{cplot},
cplot =
ContourPlot[(3*x + y Exp[x*y])*(x \[Minus] a) + (6*y +
x*Exp[x*y])*(y \[Minus] b) == 0, {x, -10, 10}, {y, -10, 10},
ContourShading -> False, Frame -> None]
]
```

It just gives a picture but i guess those better versed in Mathematica's Graphics might be able to extract at True/False therefrom. It would of course not be a guaranteed resutl, since plotting uses numeric approximation methods.

It gives a nice result for a=-4, b=-1.

--- edit ---

A comment asks about a specific set of inputs for {a,b}. Not one to duck such a test, I'll show a result with FindRoot. Here we find an {x,y} pair for which the expression of interest is negative (equal to -0.2), by setting y first to 0. I did this because the contour plot indicated there was a negative region in that general vicinity.

```
In[339]:= FindRoot[((3*x + y Exp[x*y])*(x - a) + (6*y +
x*Exp[x*y])*(y - b) /. {a -> -1.0643, b -> -.15,
y -> 0.}) == -.2, {x, .1}]
Out[339]= {x -> -0.0634401}
```

--- end edit ---

Is the set Finite? If so use the function FindInstance or use Solve or NSolve:

http://reference.wolfram.com/mathematica/ref/FindInstance.html?q=FindInstance&lang=en

If the set is not finite, then please clarify what you mean by "finding the set".

@Searke: I have no idea if the set is finite, or empty. That is part of my questions too. – Tim – 2012-03-13T21:48:19.120

Can you describe what you would like to do as you would do it by hand on a chalkboard or with a different piece of software? – Searke – 2012-03-13T21:53:58.847

If you are asking me how I do it by hand, I can tell you I will be stuck. If you are asking me why I want to know the result, my part 2 explains that, I think. – Tim – 2012-03-13T21:59:29.853

Please also include the expressions in correct Mathematica syntax. It is appreciated if you include formatted math for readability, but it is important to also have directly copyable code. – Szabolcs – 2012-03-14T05:59:19.947