Is 3 less than or equal to 5?

Many students are confused by the claims that

x<y \Rightarrow x \leq y


x<0 \Rightarrow x \leq 0.

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 3 \leq 5 ? How confident would they be about their answer?



2 responses to “Is 3 less than or equal to 5?

  1. It looks like this very concrete question may be OK after all:
    I have just received a report that our first years resoundingly agree that 3 really is less than or equal to 5.
    So presumably the problems begin with the more abstract setting?


  2. See also my more recent post on this at
    Even the concrete questions cause problems. Every year students ask me why 2 \leq 3 and/or why 1 \leq 1.


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