Many students are confused by the claims that
Indeed some refuse outright to accept that these statements are correct.
However, the same students may well have little trouble understanding about intervals in the real line, and be able to sketch open intervals and closed intervals. So perhaps once you look at “one-dimensional geometry” the problems go away. These sketches (using leftness and rightness to get a physical feel for inequalities) may be the best way to illustrate these concepts.
As for the title of this post: what would your students say if you asked them whether or not ? How confident would they be about their answer?