Question

What is the area enclosed by `2|x|+3|y|<=6`?

Answer 1

`A = 12`

Explanation:

The absolute value is given by

`|a| = {(a, a >0),(-a,a<0) :}`

As such, there will be four cases to consider here. The area enclosed by `2|x|+3|y|<=6` is going to be the area enclosed by the four different cases. These are, respectively:

`diamond x >0 and y>0`

`2|x|+3|y|<=6`
`2x+3y<=6 => y<=2-2/3x`

The portion of the area we seek is going to be the area defined by the graph

`y = 2-2/3x`

and the axes:

Since this is a right triangle with vertices `(0,2)`, `(3,0)` and `(0,0)`, its legs will have lenghts `2` and `3` and its area will be:

`A_1= (2*3)/2=3`

The second case is going to be

`diamond x < 0 and y > 0`

`2|x|+3|y| <=6`
`-2x+3y <= 6 => y<=2+2/3x`

Again, the needed area is going to be defined by the graph `y=2+2/3x` and the axes:

This one has vertices `(0,2)`, `(-3,0)` and `(0,0)`, once again having legs of lenght `2` and `3`.

`A_2 = (2*3)/2 = 3`

There is clearly some sort of symmetry here. Analogously, solving for the four areas will yield the same result; all triangles have area `3`. As such, the area enclosed by

`2|x| + 3|y| <=6`

is

`A=A_1+A_2+A_3+A_4=4*3 = 12`

As seen above, the shape described by `2|x|+3|y|<=6` is a rhombus.