Linear Inequalities and Feasible Regions: Quick Revision

A brief note on Linear Inequalities and Feasible Regions. Read it before a test. Try each question at the end before you open its answer.

Graphing an inequality

• The boundary line: replace the sign by $=$. Draw the line from its intercepts, using $y = 0$ and $x = 0$.
• The test point: try a point that is not on the line, usually the origin. Shade its side if the test is true, and the other side if it is false.
• The line style: solid for $\le$ and $\ge$. Dashed for $<$ and $>$.
• A line through the origin: test another point, such as $(1, 0)$ or $(0, 1)$.
• Dividing by a negative number: turn the sign round.

The feasible region

• Feasible solution: a point that obeys every constraint.
• Feasible region: all feasible solutions together. Shade each inequality and keep the overlap.
• Non-negativity: $x \ge 0$ and $y \ge 0$ keep the region in the first quadrant.
• Bounded: a circle can enclose it. Unbounded: it runs on for ever.
• Empty: contradictory rules leave no point, so there is no feasible solution.
• Convex: the line joining any two of its points stays inside.

Corner points

• A corner: a feasible point where two boundary lines meet.
• To find them: list all the boundary lines including the axes. Solve each pair. Keep a point only if it obeys every inequality.
• Redundant rule: a rule that changes none of the corners.

A worked example

Find the corner points of $x + y \le 7$, $x + 3y \le 15$, $x \ge 0$, $y \ge 0$.

1. The lines. $x + y = 7$, $x + 3y = 15$, $x = 0$ and $y = 0$. That gives $6$ pairs.
2. The pairs. The first two meet where $2y = 8$, so $y = 4$ and $x = 3$. The first line meets the axes at $(7, 0)$ and $(0, 7)$. The second meets them at $(15, 0)$ and $(0, 5)$. The axes meet at $(0, 0)$.
3. The tests. $(0, 7)$ gives $0 + 21 = 21 > 15$, so it fails. $(15, 0)$ gives $15 > 7$, so it fails. The other four pass every rule.

The corners are $(0, 0)$, $(7, 0)$, $(3, 4)$ and $(0, 5)$.

Practice with new numbers

A program to try

The program finds the corner points of the region in the worked example and tests two points against it.

Mistakes to avoid

• Shading without a test point.
• A solid line for a strict sign.
• Leaving out $x = 0$ and $y = 0$ when listing the boundary lines.
• Keeping a meeting point that breaks a rule.
• Not turning the sign when dividing by a negative number.

Check yourself

1. Solve $5 - 3x < 14$.

2. Where does $4x + 3y = 24$ cut the axes?

3. Is $(2, 2)$ in the region $x + y \le 7$, $x + 3y \le 15$?

4. Is the region $x \ge 0$, $y \ge 0$, $x + 2y \ge 6$ bounded?

5. How many pairs of boundary lines must you solve for a region with four constraints and the two axes?

Sources

• NCERT, Mathematics Textbook for Class XII, Part II, Chapter 12, Linear Programming.
• NCERT, Mathematics Textbook for Class XI, the chapter on Linear Inequalities.