I'm having trouble coming up with a reduction for the integer programming problem when the variables aren't constrained to $0$ or $1$. For the case where the variables are constrained to $0$ or $1$, we can use 3SAT to show that IntProg is $NP$-complete. We simply have a clause for each equation which is in the form of $x1 + x2 + x3 <= 1$
If we allow the variables to be just any integer, then could a valid way to use 3SAT be to map integers $<= 0$ to $0$, and $>= 1$ to $1$? Or Is there another reduction I am overlooking?