Find the area of the region formed by the points satisfying
`|x| + |y| + |x+y| le 2.`

Answer

3 sq. units

Explanation

`"If" x,y gt 0,` then we have
`x+y+x+y le2 " or " x+y le 1`
These inequalities form a triangle region in the first quadrant formed by axes and lines x+y = 1.

`"If" x,y lt 0,` then we have
`-x-y-x-y le2 " or " x+y ge -1`
These inequalities form a triangular region in the third quadrant formed by axes and lines x+y=-1.

`"If" x gt 0,y lt 0 " and " x+y gt 0,` then we have
These inequalities form a triangular region in the fourth quadrant formed by a-axis, a+xy = 0 and x =1.

`"If" x gt 0,y lt 0 " and " x+y lt 0,` then we have
`x-y-x-y le2 " or " y ge-1`
These inequalities form a trianglular region in the fourth quadrant formed by y-axis, x+y = 0 and y=-1

`"If" x lt 0,y gt 0 " and " x+y gt 0,` then we have
`-x+y+x+y le 2 " or " y le1`
These inequalities form a triangular region in the second quadrant formed by y-axis, x+y = 0 and y=1.

`"If" x lt 0,y gt 0 " and " x+y lt 0,` then we have
`-x+y-x-y le 2 " or " x ge -1`
These inequalities form a triangular region in the second quadrant formed by x-axis, x+y=0 and x =-1.

Combining all the regions, we have following region in the plane

Area of the above region `= 3 xx " Area of square having side length 1 unit."`
= 3 sq. unit.