I'm sure the answer to my question here already exists, but I'm having trouble finding a clear answer using proper formal notation instead of plain English.
I am aware of the notation used to place restrictions upon the domain of a function, i.e. excluding $x \in \left( -1, 1 \right)$ from the set of solutions to the real function $f \left( x \right) = x^2$ can be written as:
$$\left. f \right| _{ \mathbb{R} \setminus \left( -1, 1 \right) } \left( x \right) = x^2$$
However, if I have an equation/inequality (not a function) of multiple variables, and I wish to specify an additional restriction on the value of one (or more) of the variables to exclude certain solutions, what is the proper notation to use without simply writing the restriction out in plain English?
For instance, if I want to specify the part of the unit ball $x^2 + y^2 + z^2 \leq 1$ constrained between the planes $x = -\frac{1}{2}$ and $x=\frac{1}{4}$, how would I write this inequality without writing something like this? $$x^2 + y^2 + z^2 \leq 1 \text{ where } -\frac{1}{2} < x < \frac{1}{4}$$