Systems of inequalities and the feasible region

What you will be able to do

• shade the region that satisfies several inequalities at once;
• say what a feasible solution and the feasible region are;
• tell a bounded region from an unbounded one;
• recognise rules that leave no feasible solution.

All the rules together

A bakery has one helper and one oven. The helper works $8$ hours. A tray of rolls takes $1$ hour and a tray of buns takes $2$ hours. The oven holds $6$ trays in all. Let $x$ be the trays of rolls and $y$ the trays of buns. The rules are

The numbers $x$ and $y$ are the decision variables. The rules are the constraints. A point $(x, y)$ is a feasible solution if it obeys every constraint. The set of all feasible solutions is the feasible region.

To find it, draw each boundary line and shade each allowed side. Then keep only the part that every shading covers.

1. Rule 1. The line $x + y = 6$ passes through $(6, 0)$ and $(0, 6)$. The origin gives $0 \le 6$, so shade towards the origin.
2. Rule 2. The line $x + 2y = 8$ passes through $(8, 0)$ and $(0, 4)$. The origin gives $0 \le 8$, so shade towards the origin.
3. Rules 3 and 4. Keep the first quadrant.
4. Overlap. The two lines meet where $y = 8 - 6 = 2$ and $x = 4$. The overlap is a four-sided region with corners $(0, 0)$, $(6, 0)$, $(4, 2)$ and $(0, 4)$.

The point $(4, 2)$ is feasible, since $4 + 2 = 6$ and $4 + 4 = 8$. The point $(5, 2)$ is not, because $5 + 2 = 7$ is more than $6$. Points inside the shape are feasible, and so are points on its edges.

Bounded and unbounded regions

A feasible region is bounded if you can draw a circle round all of it. It is unbounded if it runs on for ever in some direction.

The bakery region is bounded, because both rules cap the trays. Now think of a diet. A person needs at least $4$ units of one vitamin, which gives $x + y \ge 4$, and at least $6$ units of another, which gives $x + 3y \ge 6$. Nothing stops the person from eating more. The region has no upper edge, so it is unbounded.

Rules with $\le$ and positive numbers keep a region near the origin. Rules with $\ge$ push it away from the origin without limit.

A feasible region is always convex. This means that the straight line joining any two of its points stays inside it. That is why its edges are straight and its corners point outwards.

When there is no feasible solution

Sometimes the rules fight each other. Take $x + y \le 2$ and $x + y \ge 5$. The two boundary lines are parallel. The first shading lies on one side of both, and the second on the far side, so they never overlap. The feasible region is empty.

Or take $x + 2y \le 4$ with $x \ge 5$. Since $x + 2y \le 4$ forces $x \le 4$ when $y \ge 0$, no point can have $x \ge 5$. Again nothing is left.

An empty region means no feasible solution. In a real problem it says that the demands cannot all be met together.

Practise with new numbers

Try it in Python

The program counts the whole-number points that satisfy the bakery rules. It plots them and the edge of the region.

Common mistakes

• Keeping the union. The feasible region is where all the shadings overlap, not the area covered by any one of them.
• Forgetting $x \ge 0$ and $y \ge 0$. Without them the region spills into the other quadrants.
• Checking a point against one rule only. A point must pass every rule.
• Reading an edge as outside. With $\le$ and $\ge$ the edges belong to the region.

Remember these

• A feasible solution obeys every constraint. All of them together make the feasible region.
• Shade each inequality and keep the overlap. The region is convex.
• Bounded means it fits inside a circle. Unbounded means it runs on for ever.
• Contradictory rules leave an empty region, so there is no feasible solution.

References

• NCERT, Mathematics Textbook for Class XII, Part II, Chapter 12, Linear Programming (feasible region, bounded and unbounded).
• NCERT, Mathematics Textbook for Class XI, the chapter on Linear Inequalities (solving a system of inequalities graphically).

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Is $(3, 3)$ a feasible solution of $x + y \le 7$, $2x + y \le 8$, $x \ge 0$, $y \ge 0$?

2. A feasible region is the inside of a four-sided shape. Is it bounded?

3. What is the feasible region of $x + y \ge 6$ and $x + y \le 4$?

4. Where do the boundary lines $x + y = 10$ and $x - y = 2$ meet?

5. Why is the region $x \ge 0$, $y \ge 0$, $x + y \ge 3$ unbounded?