Answer
See the images below.
Work Step by Step
(a)
$|x|\leq 1$ has two possible solution, due to the reason, that the value of $x$ can be either positive or negative (In case of $0$ it doesn't affect anything). So we have to solve $2$ possible inequalities:
$x\leq 1$ or $-x\leq 1$
$x\leq 1$
$x\geq -1$
The value of $x$ varies between $-1$ and $1$, so the region of solution is closed interval between lines $x=-1$ and $x=1$ (Blue Region). See the image above.
$|y|\leq 3$ Can be solved using the same method used above.
$y\leq 3$ or $-y\leq 3$
We have:
$y\leq 3$
$y\geq -3$
The solution lies between $y=3$ and $y=-3$ lines of closed interval (Green Region). See the image above.
(b) For the inequality ($xy>0$) to be true, both $x$ and $y$ have to be either positive or negative. Which means, that we need regions, where both $x$ and $y$ are either positive or negative at the same time. These regions are $1^{st}$ and $3^{rd}$ quadrants. See the image below.
