Answer
$$\eqalign{
& {\text{All the points on or inside the sphere centered at }}\left( {4,7,9} \right) \cr
& {\text{with radius 15}} \cr} $$
Work Step by Step
$$\eqalign{
& {x^2} + {y^2} + {z^2} - 8x - 14y - 18z \leqslant 79 \cr
& {\text{Group terms}} \cr
& \left( {{x^2} - 8x} \right) + \left( {{y^2} - 14y} \right) + \left( {{z^2} - 18z} \right) \leqslant 79 \cr
& {\text{Complete the square}} \cr
& \left( {{x^2} - 8x + 16} \right) + \left( {{y^2} - 14y + 49} \right) + \left( {{z^2} - 18z + 81} \right) \leqslant 79 + 146 \cr
& {\left( {x - 4} \right)^2} + {\left( {y - 7} \right)^2} + {\left( {z - 9} \right)^2} \leqslant {\left( {15} \right)^2} \cr
& {\text{The equation of the form }}{\left( {x - a} \right)^2} + {\left( {y - b} \right)^2} + {\left( {z - c} \right)^2} = {r^2} \cr
& {\text{Represents a sphere centered at }}\left( {a,b,c} \right){\text{ with radius }}r,{\text{ then}} \cr
& {\left( {x - 4} \right)^2} + {\left( {y - 7} \right)^2} + {\left( {z - 9} \right)^2} = {\left( {15} \right)^2} \cr
& {\text{Is a sphere centered at }}\left( {4,7,9} \right){\text{ with radius }}15 \cr
& {\text{Then, }}{\left( {x - 4} \right)^2} + {\left( {y - 7} \right)^2} + {\left( {z - 9} \right)^2} \leqslant {\left( {15} \right)^2}{\text{ represents}} \cr
& {\text{All the points on or inside the sphere centered at }}\left( {4,7,9} \right) \cr
& {\text{with radius 15}} \cr} $$
