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Question:

The area enclosed by 2|x| + 3|y| ≤ 6  is ____________ sq. units.

Explanation:

Consider, 2|x| + 3|y| = 6

Case 1: x > 0 and y > 0 [Quandrant I]
⇒ 2x + 3y = 6

Case 2: x < 0 and y > 0 [Quandrant II]
⇒ - 2x + 3y = 6

Case 3: x < 0 and y < 0 [Quandrant III]
⇒ - 2x - 3y = 6

Case 4: x > 0 and y < 0 [Quandrant IV]
⇒ 2x - 3y = 6

Each of these 4 lines can be drawn in their respective quadrants as shown in the figure.

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We need the area of region bounded by these 4 lines.

Area of region I = 1/2 × 3 × 2 = 3 sq. units

Similarly area of region II, III and IV = 3 sq. units each.

∴ Area of all the 4 regions combined = 3 + 3 + 3 + 3 = 12 sq. units

Hence, 12..