Question

Determine the location of a point \((x, y, z)\) that satisfies the condition(s). \(x y z<0\)

Short answer

The location of a point \((x, y, z)\) that satisfies the condition \(x y z<0\) would be in the either of the following regions: (i) \(x>0, y>0, z<0\), (ii) \(x>0, y<0, z>0\), (iii) \(x<0, y>0, z>0\), and (iv) \(x<0, y<0, z<0\).

Step-by-step solution

Interpret the inequality

The inequality x*y*z<0 has a negative result. Thus, in the multiplication of x, y, and z, one or three of them must be negative. This is because the multiplication of an odd number of negative numbers results in a negative product.

Determine the possible regions

There are four different regions that can have a negative product: (i) when x>0, y>0, z<0, (ii) when x>0, y<0, z>0, (iii) when x<0, y>0, z>0, and (iv) when x<0, y<0, z<0. These areas are, respectively: the area behind the x-y plane, the area below the x-z plane, the area to the left of the y-z plane, and the area within the octant formed by all three planes.

Formulate the final answer

The location of a point (x, y, z) that fulfills the condition x*y*z < 0 would be in either of these four regions: (i) x>0, y>0, z<0, (ii) x>0, y<0, z>0, (iii) x<0, y>0, z>0, and (iv) x<0, y<0, z<0.