(1) I think you have in mind -1, not 1.

(2) The subject header is less than useful (to the point where I very nearly skipped over the note).

(3) This is not a bug.

As for why this behavior is as designed, the denominator and numerator have a common factor of x+1, and Mathematica treats that as a removable singularity (because it is). Wolfram|Alpha takes that more pedantic approach of restricting the domain of definition to exclude singularities, whether or not removable. Both make sense, I think.

fyi;

Maple 18.01 agrees with Wolfram/Alpha on this one:

This might work for you:

And[Reduce[(-4 x^2 + 8 x + 12)/(-x - 1) < 0, x] , 
  ReleaseHold[
   WolframAlpha[
    "Domain (-4 x^2+8 x+12)/(-x-1)", {{"Result", 1}, 
     "Output"}]]] // Simplify

On the Raspberry Pi with the post Mathematica 9 release this command is more elegant and does not need to invoke WA:

And[Reduce[(-4 x^2 + 8 x + 12)/(-x - 1) < 0, x] , 
  FunctionDomain[(-4 x^2 + 8 x + 12)/(-x - 1), x]] // Simplify

because on the Raspberry we have the command FunctionDomain, which is new. I believe that Reduce does not consider the domain of the underlying function: that's not a bug, that's a feature.

This one might even be a bit nicer:

Reduce[{(-4 x^2 + 8 x + 12)/(-x - 1) < 0 && 
FunctionDomain[(-4 x^2 + 8 x + 12)/(-x - 1), x]}, x] // Simplify

If you have the Raspberry, you can of course put this together into one function:

  advReduce[eqn_] := 
 Reduce[{eqn && FunctionDomain[eqn[[1, 1]], x]}, x] // Simplify

You can call that function with the equation/inequality as an argument:

    advReduce[(-4 x^2 + 8 x + 12)/(-x - 1) < 0]
(*   x < -1 || -1 < x < 3   *)

You can also write a similar function in Mathematica 9 which then uses Wolfram Alpha:

 advReduce2[eqn_] := 
 Reduce[
    eqn && ReleaseHold[
      WolframAlpha[
       "Domain" <> (eqn[[1]] // InputForm // ToString), {{"Result", 
         1}, "Output"}]]] // Simplify

With the function call

advReduce2[(-4 x^2 + 8 x + 12)/(-x - 1) < 0]

which gives:

x < -1 || -1 < x < 3

Cheers, M.

Hi, this solve my problem! thanks.

Sorry again, using quotation marks (") in the title makes this cut: Why the "Reduce" command does not completely solve the inequality