Several half-planes overlap into a feasible polygon. The lesson builds the flagship quadrilateral and then repeats the same construction with a shifted boundary. The point is that every corner comes from exact boundary intersections, not from the drawing.
highlighted = computed this step
Second boundary
Add the second constraint x plus 2y<=4. Its intercepts are (4, 0) and (0, 2). Why: each added half-plane can only remove points from the current feasible set.
x+2y=4intercepts (4,0) and (0,2)
Intersection region
The feasible set is the intersection of all the half-planes, not any one line. The two slanted boundaries meet at (4/3, 4/3). Why: a vertex appears where enough active boundaries meet at once.
2x+y=4andx+2y=4⇒(4/3,4/3)
Parallel example
The asymmetric example uses the same overlap rule. Its recomputed fractional corner is (6/5, 8/5), and its corners are (0, 0), (2, 0), (6/5, 8/5), and (0, 2). Why: changing one boundary changes the polygon, but not the rule for constructing it.
asymmetric corner (6/5,8/5)
Four corners
The recomputed corners are (0, 0), (2, 0), (4/3, 4/3), and (0, 2). Why: listing vertices is a finite way to summarize the feasible polygon.
vertices (0,0),(2,0),(4/3,4/3),(0,2)
Diagram note
Half-planes overlap into a polygon whose corners are recomputed from boundary intersections. Pixel positions are rounded for layout; every number shown is exact.