The feasible region is the set of points that satisfy every constraint at once. You find it by drawing each boundary line, deciding the side, and keeping only the overlap.
This lesson follows writing constraints from a word problem in the linear programming chapter.
How do I draw one constraint?
Turn the inequality into an equation and draw that line. For a line such as x + y = 8, use the two intercepts: (8, 0) and (0, 8). Then choose solid or dashed from the symbol and test one point to find the side.
Worked example: three constraints
Draw the region defined by x + y ≤ 8, y ≥ 2 and y ≤ 2x, with x and y not negative.
- x + y = 8 is solid. Test (0, 0): 0 ≤ 8 is true, so shade the side containing the origin.
- y = 2 is a solid horizontal line. Test (0, 5): 5 ≥ 2 is true, so keep the side above it.
- y = 2x is solid and passes through the origin. You cannot test (0, 0) here, so test (1, 5): 5 ≤ 2 is false, which tells you the region is on the lower side, below the line.
The overlap is a triangle. Its vertices come from pairs of lines:
- y = 2 and y = 2x meet at x = 1, giving (1, 2).
- y = 2 and x + y = 8 meet at x = 6, giving (6, 2).
- y = 2x and x + y = 8 meet where 3x = 8, giving (8/3, 16/3).
Check that (3, 3) lies inside: 6 ≤ 8, 3 ≥ 2 and 3 ≤ 6 all hold.
The mistake that costs marks
When a line passes through the origin, it is tempting to test (0, 0) anyway. Substituting into y ≤ 2x gives 0 ≤ 0, which is true only because the point sits on the line, so it tells you nothing about the side.
Pick any point clearly off the line. Use (1, 5) or (5, 1), and work out whether it satisfies the inequality.
Dashed lines and strict inequalities
If the question says “more than 3 posters”, then y > 3 and the boundary line is dashed. The vertices still help you draw the region, but points on the dashed boundary are not allowed.
A point on a dashed line is not allowed, so a best answer at that vertex cannot be used as it stands. Read the question for whole-number conditions, because the nearest whole-number point inside the region may be the answer.
Check yourself
Find the vertices of the region x + y ≥ 4, x ≤ 5 and y ≤ 3, with x and y not negative.
Answer
Draw x + y = 4 (intercepts (4, 0) and (0, 4)), x = 5 and y = 3. Test (0, 0) in x + y ≥ 4: 0 ≥ 4 is false, so shade above the line.
The vertices are where the boundaries meet inside the region:
- x + y = 4 and y = 0: (4, 0).
- y = 0 and x = 5: (5, 0).
- x = 5 and y = 3: (5, 3).
- y = 3 and x + y = 4: (1, 3).
The point (0, 4) breaks y ≤ 3, so it is not a vertex.
What to study next
Go to testing an objective function at vertices to use these corners. Keep a record of any shading slips in the mistake log.
If you want a teacher to check your region line by line, see online one-to-one Additional Mathematics tuition.