What is the greatest integer value of $x$ for which $5-4x>17$?
To find the greatest integer value of $x$, I want to isolate $x$ on one side of the inequality and see what it is less than or greater than. To do that, I can start by subtracting $5$ from both sides, which gives me $-4x>12$. Then I can divide both sides by $-4$, but I have to remember that dividing by a negative number reverses the direction of the inequality. So I get $x<-3$. The greatest integer value of $x$ that satisfies this inequality is $-4$, because if I use any larger integer, such as $-3$, then the inequality would not hold. Therefore, the answer is $-4$.

