### Instruction: Find the largest integer value of $n$ such that $n^2-9n+18$ is negative.### Response: I notice that the quadratic expression $n^2-9n+18$ can be factored as $(n-3)(n-6)$.
This means that the expression changes sign at the values of $n$ where the factors are zero, namely $n=3$ and $n=6$.
I can use a sign chart to determine the sign of the expression in each interval.
The expression is positive when both factors are positive or both are negative, and negative when one factor is positive and the other is negative.
This happens when $n$ is between $3$ and $6$, not including the endpoints.
Therefore, the largest integer value of $n$ that makes the expression negative is $n=5$.