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Upgrading from:

Algebra`AlgebraicInequalities`

New function SemialgebraicComponentInstances has been added to the built-in Mathematica kernel.

The output is returned as a list of rules:

Version 5.2 << Algebra`AlgebraicInequalities`; SemialgebraicComponents[{x (x^2 - 2) (x^2 - 3) > 0}, x]
Wolfram Language code: SemialgebraicComponentInstances[{x (x ^ 2 - 2) (x ^ 2 - 3) > 0}, x]

SemialgebraicComponentInstances accepts arbitrary Boolean combinations of equations and inequalities:

Version 5.2 SemialgebraicComponents[ (x + 1/2)^2 + y^2 < 1 && (x - 1/2)^2 + y^2 == 1, {x, y}] returns unevaluated
Wolfram Language code: SemialgebraicComponentInstances[ (x + (1/2))^2 + y^2 < 1 && (x - (1/2))^2 + y^2 == 1, {x, y}]

Entering the equations as a List is equivalent to using the Boolean operator And:

Version 5.2 SemialgebraicComponents[{ (x + 1/2)^2 + y^2 < 1, (x - 1/2)^2 + y^2 == 1}, {x, y}] returns unevaluated
Wolfram Language code: SemialgebraicComponentInstances[{ (x + (1/2))^2 + y^2 < 1, (x - (1/2))^2 + y^2 == 1}, {x, y}]

You may also use other Boolean operators:

Version 5.2 SemialgebraicComponents[((x + 1/2)^2 + y^2 < 1 && (x - 1/2)^2 + y^2 == 1) || ((x + 1/2)^2 + y^2 == 1 && (x - 1/2)^2 + y^2 >= 1), {x, y}] returns unevaluated
Wolfram Language code: SemialgebraicComponentInstances[((x + (1/2))^2 + y^2 < 1 && (x - (1/2))^2 + y^2 == 1) || ((x + (1/2))^2 + y^2 == 1 && (x - (1/2))^2 + y^2 ≥ 1), {x, y}]