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

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:

![https://www.desmos.com/calculator]()

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:

![https://www.desmos.com/calculator]()

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#

![https://www.desmos.com/calculator]()

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