Answer
$$\eqalign{
& {\text{All the points on or inside the sphere centered at }}\left( {0,7,0} \right) \cr
& {\text{with radius 6}} \cr} $$
Work Step by Step
$$\eqalign{
& {x^2} + {y^2} - 14y + {z^2} \leqslant - 13 \cr
& {\text{Group terms}} \cr
& {x^2} + \left( {{y^2} - 14y} \right) + {z^2} \leqslant - 13 \cr
& {\text{Complete the square}} \cr
& {x^2} + \left( {{y^2} - 14y + 49} \right) + {z^2} \leqslant - 13 + 49 \cr
& {x^2} + {\left( {y - 7} \right)^2} + {z^2} \leqslant {\left( 6 \right)^2} \cr
& {\text{The equation of the form }}{\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} + {\left( {z - c} \right)^2} = {r^2} \cr
& {\text{Represents a sphere centered at }}\left( {a,b,c} \right){\text{ with radius }}r,{\text{ then}} \cr
& {x^2} + {\left( {y - 7} \right)^2} + {z^2} = {\left( 6 \right)^2} \cr
& {\text{Is a sphere centered at }}\left( {0,7,0} \right){\text{ with radius }}6 \cr
& {\text{Then, }}{x^2} + {\left( {y - 7} \right)^2} + {z^2} \leqslant {\left( 6 \right)^2}{\text{ represents}} \cr
& {\text{All the points on or inside the sphere centered at }}\left( {0,7,0} \right) \cr
& {\text{with radius 6}} \cr} $$
