### Instruction: What is the greatest integer $n$ such that $n^2 - 11n +24 \leq 0$?### Response: I need to find the values of $n$ that make the quadratic expression $n^2 - 11n +24$ non-positive.
One way to do that is to factor the expression and see when it is zero or negative.
I notice that $n^2 - 11n +24$ can be factored as $(n-3)(n-8)$.
This means that the expression is zero when $n=3$ or $n=8$, and it changes sign at these values.
To see when it is negative, I can pick a test value in each interval between the roots and plug it into the expression.
For example, when $n=0$, the expression is $(0-3)(0-8) = 24 > 0$, so it is positive in the interval $(-\infty, 3)$.
When $n=5$, the expression is $(5-3)(5-8) = -6 < 0$, so it is negative in the interval $(3, 8)$.
When $n=10$, the expression is $(10-3)(10-8) = 14 > 0$, so it is positive in the interval $(8, \infty)$.
Therefore, the expression is negative or zero only in the interval $[3, 8]$.
The greatest integer in this interval is $8$, so that is the answer.