Question

Describe with a sketch the sets of points \((x, y, z)\) satisfying the following equations. $$x^{2} y^{2} z^{2}>0$$

Short answer

Question: Sketch the sets of points (x, y, z) that satisfy the equation x²y²z² > 0. Answer: The sets of points that satisfy the equation x²y²z² > 0 can be represented by the union of four non-adjacent octants: the first octant (positive x, y, z axes), the octant in the negative y and negative z plane but positive x-plane, the octant in the negative x and negative z plane but positive y-plane, and the octant in the negative x and negative y plane but positive z-plane.

Step-by-step solution

Analyze the equation

Recall the inequality \(x^{2} y^{2} z^{2}>0\). We can break it down by the fact that \(x^{2}\), \(y^{2}\), and \(z^{2}\) are all non-negative. - If either \(x\), \(y\), or \(z\) is equal to 0, then their square will be equal to 0 and the product \(x^{2} y^{2} z^{2}\) will also be 0. - If all three variables are positive then their squares will be positive and the product will be positive. - If all three variables are negative then their squares will be positive and the product will be positive. - We can also have two variables being negative, then their squares will be positive and the product of the three squares will still be positive. This inequality can be satisfied if all of the variables are positive or two of them are negative. In other words, we need to show the sets of points (\(x\),\(y\),\(z\)) where x*y*z > 0.

Define regions based on sign changes

From the analysis above we can now create a table with regions where the inequality \(x^{2} y^{2} z^{2}>0\) holds true: 1. x > 0, y > 0, z > 0 2. x > 0, y < 0, z < 0 3. x < 0, y > 0, z < 0 4. x < 0, y < 0, z > 0

Sketch the graph

To create the 3D graph, we want to illustrate the four regions in which the inequality holds: 1. x > 0, y > 0, z > 0: The first octant (positive x,y,z axes) 2. x > 0, y < 0, z < 0: The octant in the negative y and negative z plane but positive x-plane. 3. x < 0, y > 0, z < 0: The octant in the negative x and negative z plane but positive y-plane. 4. x < 0, y < 0, z > 0: The octant in the negative x and negative y plane but positive z-plane. Thus, the graph representing the sets of points where \(x^{2} y^{2} z^{2}>0\) is formed by combining these four non-adjacent octants.