If the square root of w is an integer, how many values of w are between 20 and 120

So, we'll use a triplet, keeping in mind that w > 20 and that w < 120, so "w" is really greater than 20 and less than 120, but is neither, so then 20 < w < 120, there's our triplet.

now, in a triplet, just like in an equation, if you add or multiply to one side, you'd do it for all sides, in this case, 3 sides,



so, whatever the √w is, we know is between those numbers, we also know is an integer, so it doesn't have any decimal parts, so it has to be 5 or 6 or 7 or 8 or 9 or 10.  So 4 < √w < 11, or √w whatever that may be, is between 4 and 11, is neither 4 or 11, but just between.