What is the greatest integer value of $x$ for which $5-4x>17$?
First we isolate $x$ by subtracting 5 from both sides.  This gives \[-4x>12.\]Dividing by $-4$ and remembering to reverse the inequality gives us  \[x<-3.\]The greatest integer that solves this inequality is $\boxed{-4}$.

We can check this.  If we substitute $-4$ into the inequality we get  \[5-4(-4)>17\]or  \[5+16>17.\]This is true.  If we substitute $-3$ we get  \[5+12>17,\]which is false.
-4
Level 4
Prealgebra
Problem: What is the greatest integer value of $x$ for which $5-4x>17$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the inequality \(5 - 4x > 17\), first isolate the term involving \(x\). Subtract 5 from both sides to obtain \(-4x > 12\). Next, divide both sides by -4. Remember that dividing by a negative number reverses the inequality sign, leading to \(x < -3\). The greatest integer value satisfying this condition is \(-4\). Thus, the final answer is \(-4\).<|END_OF_LATENT|> The final answer is $-4$.