Answer
The objective function has maximum value 12
at (0,4) and (3,2)
Work Step by Step
The region that is defined by the constraints is
* to the right of x=0 (the y=axis),
* above the line y=0 (the x-axis),
* below the line 2x+y=8,
(for this line, when x=0, y=8 and when y=0, x=4)
* below the line 2x+3y=12
(for this line, when x=0, y=4 and when y=0, x=6)
The region is shaded red on the image below.
The corners are (0,0), (4,0), (3, 2) and (0,8).
Calculating the objective function z=2x+3y in each corner point,
(table in step 6 of the image)
we find that
the objective function has maximum value 12
at (0,4) and (3,2)