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Algebra`AlgebraicInequalities`
The package provides a function for solving systems of strong polynomial inequalities in one or more unknowns. To be precise, SemialgebraicComponents[ineqs,vars] gives a finite set of solutions of the system of inequalities, such that any other solution can be connected by a continuous path of solutions with one of the solutions in the set. ineqs is a list of strong inequalities, where both sides of each inequality are polynomials in variables vars with rational coefficients. In other words, SemialgebraicComponents[ineqs,vars] gives at least one point in each connected component of the open semialgebraic set defined by inequalities ineqs.
Solutions of inequalities.
This loads the package.
In[1]:= <<Algebra`AlgebraicInequalities`
Here is a point from each of the three intervals forming the set of solutions of 
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In[2]:= SemialgebraicComponents[{x (x^2 - 2) (x^2 - 3) > 0}, x]
Out[2]= 
This gives one point in each of the two connected components of the set bounded by the circle  and the hyperbola 
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In[3]:= SemialgebraicComponents[{x^2 + y^2 < 4, x y > 1}, {x, y}]
Out[3]= 
This proves that the ball  is contained in the ellipsoid 
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In[4]:= SemialgebraicComponents[{x^2 + y^2/4 + z^2/9 > 1,
x^2 + (y - 1)^2 + (z - 2)^2 < 1/9},
{x, y, z}]
Out[4]= 
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