Give a geometric description of the following set of points. \( x^{2}-12 x+y^{2}+z^{2} \geq-20 \) Choose the correct answer below. A. The exterior of a ball with center \( (6,0,0) \) and radius 4 (including the sphere itself) B. A ball with center \( (-6,0,0) \) and radius 4 C. A ball with center \( (6,0,0) \) and radius 4 D. The exterior of a ball with center \( (-6,0,0) \) and radius 4 (including the sphere itself) E. The entire xyz-coordinate system F. The empty xyz-coordinate system

To analyze the inequality \( x^{2}-12 x+y^{2}+z^{2} \geq-20 \), we first rearrange the equation. Completing the square for the \( x \) terms gives: \[ (x-6)^{2} - 36 + y^{2} + z^{2} \geq -20 \] This simplifies to: \[ (x-6)^{2} + y^{2} + z^{2} \geq 16 \] The right side can be expressed as \( 4^2 \), which indicates a relationship to a sphere. This inequality describes the exterior of a sphere (including its surface) centered at \( (6, 0, 0) \) with a radius of 4. Therefore, the correct answer is: **A. The exterior of a ball with center \( (6,0,0) \) and radius 4 (including the sphere itself)**. Now for a little bit more insight: In mathematical terms, the inequality illustrates points that are at a distance greater than or equal to 4 units from the point (6, 0, 0). This means any point within that distance (inside the ball) does not satisfy the inequality, leading to a fun visual of the space outside of that spherical ball while allowing the surface itself! If you enjoy geometry, visualizing 3D shapes can be quite fun! Imagine blowing up a balloon from the center point (6, 0, 0) outwards; all those points on the balloon and outside are included in this set. It’s a fantastic way to see mathematics in action!