Just had my first lecture today and was working on some practice problems, but got stuck on this one. All I have been introduced to so far is the equation of a sphere, but I don't really know what else to do with this.
I completed the square on both sides of the inequality by adding 1/4 to each side and made the middle of the inequality a sphere centered at (0,0,1/2).
Here is my full attempt. The inequality represents the region on/between the spheres centered at (0,0,1/2) with radii 1 1/4 and 3 1/4. The region contains all of the points on the sphere with radius 1 1/4 and includes all of the points of the second sphere except the outermost points where r = 3 1/4.
I think this might be right, but I just thought I'd check with you guys.
dwsmith, I think you must have misunderstood the question.
statelyplump, in addition to what Soroban said, you are misusing some words. Strictly, a "sphere" is the surface of a "ball" and, specifically, to say a point is "on a sphere" implies it is on the surface. When you say "includes all of the points of the second sphere except the outermost points", that is, at best, misleading. What you mean is that it "includes all point within the second sphere and outside the first sphere" or "includes all points inside the ball of radius and on or outside the ball of radius .