Variable intervals from system of inequalities ( − 1 0 1 0 0 − 1...

The system is actually not too complicated. If you multiply them out you should get
-i < 20
i < 20
-j < 10
i-j < 0
-2i + j < 20
From the 2nd inequality, we have our desired upper bound for i. For the lower bound of -15, combine j > -10 and -2i + j < 20.
The 3rd inequality gives us our desired lower bound for j. As for the upper bound of 60, combine i < 20 and -2i+j < 20.

$\left(\begin{array}{cc}-1& 0\\ 1& 0\\ 0& -1\\ 1& -1\\ -2& 1\end{array}\right)\left(\begin{array}{c}i\\ j\end{array}\right)<\left(\begin{array}{c}20\\ 20\\ 10\\ 0\\ 20\end{array}\right)$
$\left(\begin{array}{c}-i\\ i\\ -j\\ i-j\\ -2i+j\end{array}\right)<\left(\begin{array}{c}20\\ 20\\ 10\\ 0\\ 20\end{array}\right)$
$\left(\begin{array}{c}-i<20\\ i<20\\ -j<10\\ i<j\\ -2i+j<20\end{array}\right)$