Answer
$1\leq x^{2}+y^{2}+z^{2}\leq 4$
Work Step by Step
The $outer $ sphere:$\quad x^{2}+y^{2}+z^{2}=2^{2}$
The $inner $ sphere:$\quad x^{2}+y^{2}+z^{2}=1^{2}$
Each equation states that the square of the distance of a point on the sphere from its centerpoint EQUALS $($the radius$)^{2}$.
We want the square of the distance to be between $1^{2}$ and $2^{2},$ inclusive$:$
$1\leq x^{2}+y^{2}+z^{2}\leq 4$
