subject to inequalities are your constraints.
using the desmos.com calculator, you would graph the opposite of the inequalities.
the area on the graph that is not shaded is the region of feasibility.
the corner points of this region are where your maximum values of the objective function will be.
the corner points are in (x,y) format.
evaluate x + y using the values at each corner point.
the corner point with the greatest value of x + y is your maximum value of x + y.
the graph looks like this.
your points are at (0,6), (4,4), (6,0), (0,0)
your values of x + y at these points are 6, 8, 6, 0
if the points are allowed to be on the lines, your maximum value would be 8.
because the points can't be on the lines, your maximum value will be some value less than 8.
that's still the maximum value because all the other values are less than a maximum number that is less than 8.
fyi, i put this in a simplex modeling tool that is used to solve problems like this and it said it can't deal with <; it can only deal with <=.
i graphed the inequalities as <= or >=.
the corner points are the same as when i graphed the inequalities as < or >.
it's the evaluation at those corner points that counts.
if all inequalities were <=, then the value at the corner point is valid.
if some or all the inequalities were <, then the value at the corner point needs to be checked to see that all the inequalities are satisfied at that corner point.
the simplex modeling tool i used is at:
https://www.zweigmedia.com/RealWorld/simplex.html
the desmos.com graphing calculator that i used is at:
https://www.desmos.com/calculator
here's a display of the graphing method in the two forms described above (using <= and using <) to show you that the corner points are the same.
here's a display of the results using the simplex modeling tool in the two forms describe above to show you that the simplex modeling tool could handle <= and couldn't can't handle <.
