First graph the boundary lines, one at a time, which are the lines whose
equations are just like the inequalities with the symbols of inequality changed
to symbols of equality ("=").
We graph the first boundary line (in green), x + y = 4, which has intercepts
(0,4) and (4,0)
Before we draw the other boundary line, let's find out which side of the line
all the solutions to the inequality x + y ≤ 4 are on. We do that by
substituting any point that isn't on the line into the inequality as a test
point. The easiest test point to substitute is the origin (0,0). We can use
it as a test point because (0,0) doesn't lie on the line. Substituting
(x,y) = (0,0)
x + y ≤ 4
0 + 0 ≤ 4
0 ≤ 4 <--- this is true so the solutions lie on the side of the line which
the test point (0,0), the origin, lies on, which is BELOW and to the LEFT of the
green line.
Next we graph the second boundary line (in blue), 2x - y = 4, which has
intercepts (0,-4) and (2,0)
Let's find out which side of the line all the solutions to the inequality
2x - y ≤ 4 are on. Again we do that by substituting any point that isn't on the
line into the inequality as a test point. The easiest test point to substitute,
again, is the origin (0,0). We can use it again as a test point because (0,0)
doesn't lie on the line. Substituting
(x,y) = (0,0)
2x - y ≤ 4
2(0) + 0 ≤ 4
0 ≤ 4 <--- this is true so the solutions lie on the side of the line which
the test point (0,0), the origin, lies on, which is ABOVE and to the LEFT of the
blue line.
So finally we shade the area which is BELOW and to the LEFT of the
green line AND which is ABOVE and to the LEFT of the blue line.
Edwin