10 min readPathway sheet1-mark MCQ on a described graph; the drawing step of every 3, 4 or 5-mark LPP answer

Feasible and infeasible regions (bounded or unbounded)

The feasible region is the common region of all the constraints, including x ≥ 0, y ≥ 0. It is bounded if a circle can enclose it.

Feasible regionThe common region of all the constraints, including x ≥ 0, y ≥ 0

Infeasible regionThe part of the plane outside the feasible region

Bounded regionCan be enclosed within a circle

Unbounded regionStretches without end in some direction; no circle can enclose it

How to draw the feasible region

1
Change each inequality into an equation; for example, 2x + y ≤ 12 becomes 2x + y = 12.
2
Find where each line cuts the axes: put x = 0, then y = 0. For 2x + y = 12 these are (0, 12) and (6, 0).
3
Draw each line with a ruler on the same axes, using a clear scale, and label it with its equation.
4
Test O(0, 0) in each inequality. If it is true, shade the side containing O; if false, shade the other side. (If a line passes through O, test a point such as (1, 0).)
5
Keep only the first quadrant, because x ≥ 0 and y ≥ 0.
6★
The region shaded for every constraint is the feasible region. Shade it darker, mark its corner points and write their coordinates.

★ key step

Example 1: a bounded region

Problem

Draw the feasible region for 2x + y ≤ 12, x + 2y ≤ 12, x ≥ 0, y ≥ 0.

Working

2x + y = 12 passes through (6, 0) and (0, 12). O gives 0 ≤ 12, true: shade towards O.
x + 2y = 12 passes through (12, 0) and (0, 6). O gives 0 ≤ 12, true: shade towards O.
The lines meet where 2x + y = 12 and x + 2y = 12. Subtracting gives x − y = 0, so x = y = 4.
The common region is OABC with O(0, 0), A(6, 0), B(4, 4), C(0, 6).

Answer

Feasible region OABC, a bounded quadrilateral with corners (0, 0), (6, 0), (4, 4), (0, 6).

Example 2: an unbounded region

Problem

Draw the feasible region for 2x + y ≥ 8, x + 2y ≥ 10, x ≥ 0, y ≥ 0.

Working

2x + y = 8 passes through (4, 0) and (0, 8). O gives 0 ≥ 8, false: shade away from O.
x + 2y = 10 passes through (10, 0) and (0, 5). O gives 0 ≥ 10, false: shade away from O.
The lines meet at (2, 4): from 2x + y = 8, y = 8 − 2x; then x + 16 − 4x = 10, so x = 2 and y = 4.
The region above both lines in the first quadrant has corners (0, 8), (2, 4), (10, 0).

Answer

The feasible region is unbounded, with corners (0, 8), (2, 4) and (10, 0).

Example 3: no feasible region

Problem

Draw the constraints x + y ≤ 2, 2x + 3y ≥ 12, x ≥ 0, y ≥ 0. What do you find?

Working

x + y ≤ 2 gives the triangle with corners (0, 0), (2, 0), (0, 2).
2x + 3y = 12 passes through (6, 0) and (0, 4); the region 2x + 3y ≥ 12 is away from O.
At the corners of the triangle, 2x + 3y is 0, 4 and 6, all less than 12.
So no point of the triangle satisfies 2x + 3y ≥ 12.

Answer

The constraints have no common region. The LPP has no feasible solution.

Bounded vs unbounded feasible region

BoundedClosed on all sides, like a polygon. A circle can enclose it. Z = ax + by always has both a maximum and a minimum on it.

UnboundedOpen on one side and stretches without end. A maximum or minimum may not exist; you must test the best corner value with an extra half-plane.

In real life

When a family plans the month's groceries with limits on money and storage space, every shopping plan that fits both limits is a point in the feasible region.

Exam angle

1 marks: “The feasible region of an LPP is the triangle with vertices (0, 0), (4, 0) and (0, 3). Is it bounded or unbounded?” Bounded, because a circle can enclose the triangle.

2 marks: “Draw the feasible region for x + y ≤ 5, x ≤ 3, x ≥ 0, y ≥ 0 and write its corner points.” Draw x + y = 5 (through (5, 0) and (0, 5)) and x = 3. Shade towards O for both. Corners: (0, 0), (3, 0), (3, 2), (0, 5). The region is bounded.

1 marks: “What can you say about an LPP whose constraints have no common region?” It has no feasible region, so it has no feasible solution and no optimal value.

Remember it

Test the origin, shade the truth. If O makes the inequality true, shade towards O.