Questions on the functionality of Mathematica for Boolean computation.

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3D Contour Plot parts of sphere with gradient less than 0, equal to 0 and more than 0

I want to plot only parts of the sphere of radius 1 where the gradient $\frac{\partial z}{\partial x}$ is < 0, = 0 and > 0. The full sphere is given by: ...
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Split absolute value in chambers

I usually expand expression containing absolute values, with the PiecewiseExpand[] function. ...