First slide

Cartesian plane

Question

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

Moderate

A

3 sq. units
B

4 sq. units
C

12 sq. units
D

24 sq. units

Solution

The given inequality is equivalent to the following system of inequalities.
$2 x + 3 y \leq 6$ , when $x \geq 0, y \geq 0$
$2 x - 3 y \leq 6$ , when $x \geq 0, y \leq 0$
$- 2 x + 3 y \leq 6,$ when $x \leq 0, y \geq 0$
$- 2 x - 3 y \leq 6$ , when $x \leq 0, y \leq 0$
which represents a rhombus with sides
$2 x + 3 y = 6$ and $2 x + 3 y = - 6$
Length of the diagonals is 6 and 4 units along x-axis and
y-axis.
$∴$ The required area $= \frac{1}{2} \times 6 \times 4 = 12$ sq. units.

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