The area ( in square units) enclosed by $2\left|x\right|+3\left|y\right|\le 6$ is

We have $2\left|x\right|+3\left|y\right|\le 6$ This is equivalent to the following system of inequations:

and, $\begin{array}{r}2x+3y\le 6,\text{ when }x\ge 0,y\ge 0\\ 2x-3y\le 6,\text{ when }x\ge 0,y\ge 0\\ -2x+3y\le 6,\text{ when }x\le 0,y\ge 0\\ -2x-3y\le 6,\text{ when }x\le 0,y\le 0\end{array}$

These inequations form a rhombus having its diagonals of lengths 6 and 4 units, along the coordinate axes

So, required area $=\frac{1}{2}(6\times 4)=12$ sq. units