I've noticed that my students (primarily first-year students taking their first college math course) have a lot of trouble with inequalities. Not the math, the purpose: they seem to have a preconception that math is supposed to be about finding things exactly, and every time we talk about a bound, I get multiple people asking why anyone would ever care about such a thing.
I've been sort of waving my hands and saying "Well, if you build a bridge, you care about knowing the smallest weight which might cause it to break..." but I think it might help to be more concrete. I'm wondering if anyone can suggest good techniques or examples which illustrate the importance of bounds. (Given that the students aren't mathematicians, uses outside of math which make sense even in a world with computers that produce calculations accurate to 30 decimal places are particularly useful.)