How many integers $n$ does $\sqrt{n}$ differ from 11 by less than 1? I actually found this question pretty straightforward.
My approach : $$\sqrt{n} - 11 < 1$$ so $$\sqrt{n} - 12 < 0$$ So since $n$ is a positive integer, I thought there are at least 143 values up to $\sqrt{144} = 12$ which satisfies this inequality, but when I checked the solutions, it said the answer is 43. Why? What did I do wrong?
Hint: $\sqrt{n}$ differs from $11$ by less than $1$ if and only if $$|\sqrt{n} - 11| < 1$$
Your mistake was that you only considered the integers whose square roots were less than $1$ greater than $11$ (i.e., $\sqrt{n}<12$), but forgot to look at the other side, i.e., integers whose square roots were greater than $1$ less than $11$ (i.e., $\sqrt{n} > 10$). Considering both of these together is what gives the solution.