First slide

Equation of line in Straight lines

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.
$\begin{aligned} 2 x + 3 y \leq 6, when x \geq 0, y \geq 0 \\ 2 x - 3 y \leq 6, when x \geq 0, y \leq 0 \\ - 2 x + 3 y \leq 6, when x \leq 0, y \geq 0 \\ - 2 x - 3 y \leq 6, when x \leq 0, y \leq 0 \end{aligned}$
which represents a rhombus with sides
$2 x \pm 3 y = 6 and 2 x \pm 3 y = - 6$
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) x 6 x 4 = 12 sq. units.

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