Integer

What is the greatest integer n such that \(n^2 - 11n +24 \leq 17n - 8\)?

Combining like terms: \(n^2-28n+32\leq0\)

Using quadratic formula n = \({28\pm4\sqrt{41}\over2}=14\pm2\sqrt{41}\). Since our quadratic must be nonnegative, when factored (x - r)(x - s) where r and s are our roots, they must both be positive or both negative, so we take the extreme intervals: \((-\inf,14-2\sqrt{41}]\) U \([14 + 2\sqrt{41}, \inf)\)

Borrowing from proyaop's answer

14 + 2sqrt (41)  ≈ 26.8

This is  the larger root  and all  integer x's > than this will make the polynomial > 0

So.....the greatest integer that makes  this le to 0   is 26