in the attached graph, the feasible region is the region that is NOT shaded.
here's the graph:
the corner points are:
(10,10)
(20,0)
(24,0)
(24,24)
evaluate the objective function at these corner points and the minimum cost is when 10 full time employers are hired and 10 part time employees are hired.
the objective function is the cost function of z = 18x + 12y
at the coordinate point of (10,10), the cost is 180 + 120 = 400.
the constraints at (10,10) are satisfied.
x + y >= 20
x <= 24
x >= y
the total constraints are:
x + y >= 20
x <= 24
x >= y
x >= 0
y >= 0
the last two are necessary because neither x nor y can be less than 0.
in the graph, i shaded the regions that do NOT satisfy the constraints.
what is left is the region that DOES satisfy the constraints.
you still need to pay attention to the original constraints since that tells you what your answer can be.
for example, if the constraint was x + y > 20, then (10,10) would not satisfy that constraint because it is not greater than 20.
(10,10) would still be a marker that tells you the area where the answer might lie.
(11,10) is still in the region of feasibility, as is (10,11).
either one of those would suffice if the requirement was > rather than >=.
in fact (10,11) would be the least cost solution in that case, since the pay of the temporary employee is less than the pay of the regular employee.
the software to graph that i used is at www.desmos.com
here's a reference on graphing linear inequalities.
http://www.purplemath.com/modules/ineqgrph.htm
note that they tell you to shade the region of feasibility and that to use dashed lines if the inequality does not contain an equal as part of it.
i did the opposite only because i was not manually creating the graph and found that not shading the region of feasibility allowed it to show up better when using desmos software to generate the graph.
the general idea is that you graph the region of feasibility and then look for the corner points of that region.
your minimum / maximum solution, if there is one, should be at those corner points.
here's another reference that addresses corner points.
http://www.purplemath.com/modules/linprog.htm