Question

The area enclosed by $2|x|+3|y|\le 6$ is

• A. 3 sq. units
• B. 4 sq. units
• C. 12 sq. units
• D. 24 sq. units

Answer 1

Answer: (C)

The given inequality is equivalent to the following system of inequalities.
$2x+3y\le 6,$ when $x\ge 0,y\ge 0$
$2x-3y\le 6,$ when $x\ge 0,y\le 0$
$-2x+3y\le 6,$ when $x\le 0,y\ge 0$
$-2x-3y\le 6,$ when $x\le 0,y\le 0$
which represents a rhombus with sides
$2x\pm 3y=6$ and $2x\pm 3y=-6$
The lengths of the diagonals are $6$ units and $4$ units, respectively, along the $x$ - and $y$-axes. Therefore, the required area is $1/2\times 6\times 4=12$ sq. units.