Question 1083513
The solution to each (compound) inequality can be graphed as
2 parallel lines, and all the (x,y) points in between.
The solution to the whole system is a parallelogram including its sides.
Point (0,0) is included in the parallelogram.
The lines x+y=constant are parallel to each other,
but not parallel to any side of the parallelogram,
So, one of them will just touch one vertex.
The sum x+y at that point is the maximum.
{{{drawing(300,300,-5,5,-5,5,
line(-5,5,5,-5),line(-5,8,5,-2),
locate(-4.5,3,x+y=0),locate(1,2.5,x+y=3),
graph(300,300,-5,5,-5,5,
x+4,x-4,(3-7x)/2,-(3+7x)/2))}}}
That vertex must be the intersection of boundary lines
{{{7x+2y=3}}} and {{{y=x+4}}} <---> {{{y-x=4}}}.
Substituting, we get
{{{7x+2(x+4)=3}}}
{{{7x+2x+8=3}}}
{{{9x=-5}}}
{{{x=-5/9}}}
Then,{{{y=-5/9+4=31/9}}} , and
{{{x+y=-5/9+31/9=26/9=2&8/9}}}
Just graph the lines, and check my calculations.