Find the greatest value of $a$ such that $a^2-10a+21 \le 0$.
I notice that the expression on the left is a quadratic function of $a$, and it is negative or zero when $a$ is between its two roots. I can find the roots by factoring the expression: $a^2-10a+21 = (a-3)(a-7)$. The roots are $a=3$ and $a=7$, and the function is negative or zero between them, so the greatest value of $a$ that satisfies the inequality is $a=7$.

