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I frequently have trouble trying to place limits on Solve or Reduce using assumptions and domains. For example, this code:

Assuming[x > 0 && x < 1, Solve[(1 - (1 - x)^3)^3 == x, x, Reals]] 

Produces results both with complex number, and with x outside the range specified. The exact output is long (and includes the desired results), but a subset is pasted below. What is going on? $\left\{\{x\to 0\},\{x\to 1\},\left\{x\to 1-\frac{\left(1-i \sqrt{3}\right) \left(\frac{1}{2} \left(-9+\sqrt{93}\right)\right)^{1/3}}{2 3^{2/3}}+\frac{1+i \sqrt{3}}{2^{2/3} \left(3 \left(-9+\sqrt{93}\right)\right)^{1/3}}\right\}\right\}$

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Solve[{(1 - (1 - x)^3)^3 == x, 0 < x < 1}, x, Reals] works fine in Mathematica 8... Reduce[{(1 - (1 - x)^3)^3 == x, 0 < x < 1}, x, Reals] is pretty much equivalent.

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    I would like to disagree. Your first example does NOT work fine in Mathematica 8: In[16]:= Solve[{(1 - (1 - x)^3)^3 == x, 0 < x < 1}, x, Reals] During evaluation of In[16]:= Solve::eqf: x<1 is not a well-formed equation. >>2014-02-23