Given any line in 3-space, what is the maximum and minimum number of regions that would it would pass through? Show how you know.
I am lost with this question because I know that there are $8$ regions in the $xyz$ plane but don't know the maximum or minimum. Can the minimum be $2$ regions if the line is horizontal or vertical?
I assume by "region" you mean orthant bounded by the $x=0$, $y=0$, and $z=0$ planes.
Two regions might be correct for the minimum, but there are some degenerate cases you need to think about. Which region(s) does the $x=0$ plane belong to? How many regions does one of the coordinates axes (say, the $x$ axis) pass through? A reasonable case could be made for zero.
The precise definition of "passing through a region" also matters for counting the maximum number of crossings. But let's assume a line does not lie in any of the coordinate planes. In that case, it passes from one region into another whenever it intersects the $x=0$, $y=0$, or $z=0$ plane. A line intersects each plane at most once, and so passes through at most four regions. (It might be easier to visualize the 2D case first: a line can pass through at most three quadrants, switching quadrants twice, once when it hits $x=0$ and once when it hits $y=0$.)