The number of distinct pairs of integers $(m,n)$ satisfying $|1+mn|<| m+n|<5$ is? $|1+mn|<5$
$\Rightarrow -6 < mn < 4$
$|m+n|<5$
$\Rightarrow -5<m+n<5$
Now as $m$ and $n$ are integers , I tried to make cases such the product of $mn$ is equal to $-5,-4,-3,-2,-1,0,1,2$ and $3$ as $ -6 < mn < 4$ but I end up getting too many pairs and also I am confused what to do in the case where $mn=0$.

The official solution is right: There are 12 solutions:

*

*(-4, 0)

*(-3, 0)

*(-2, 0)

*(0, -4)

*(0, -3)

*(0, -2)

*(0, 2)

*(0, 3)

*(0, 4)

*(2, 0)

*(3, 0)

*(4, 0)

