The feasible region as the overlap of the half planes the constraints allow

What to assume they know

• Graphing a linear inequality in two variables, from Class XI, including deciding which side of the line it allows
• Drawing a straight line from its equation using the two points where it meets the axes
• Solving two linear equations together to find where two lines cross
• Substituting a pair of numbers into an inequality and getting a verdict
• The formulation vocabulary of the previous module: objective function, constraints, non-negative restrictions

What they should be able to do

• Draw the line belonging to a constraint, and decide by testing one point which side of it the constraint allows
• Explain why a non-strict inequality keeps its own boundary line and a strict one does not
• Build the region allowed by several constraints as the overlap of the regions each one allows separately
• Name the region and its complement in the chapter's own vocabulary, and give both of the chapter's names for the region itself
• Decide, for a given point, whether it lies inside the region, on its boundary, or outside, and say what each verdict is called
• Apply the chapter's circle test to say whether a region is bounded, and state the test's limits
• Explain why an unbounded region need not run away in every direction, using the chapter's own unbounded example
• Recognise a system whose region is empty, say what that means for the problem, and identify it from the drawing
• State what the chapter says about convexity and be honest that the chapter proves nothing about it
• Say why the region alone does not answer the question, and what the next topic has to add

Where it usually goes wrong

• "A feasible solution is a corner point." It is any point of the region, boundary included, and the region has infinitely many. The chapter's three sample feasible points are all corners and one of them is the eventual answer, which is exactly how this belief forms. Lead with an interior point.
• "The boundary is the edge, so it is outside." In this chapter every constraint permits equality, so every boundary line belongs to the region. The chapter says so in the same breath as it defines a feasible solution.
• "Every line on a linear programming graph is solid." Every constraint line is. The chapter draws one dashed line, in Fig 12.5, and it is dashed because it comes from a strict inequality. A student who draws it solid has claimed that points on it satisfy a condition they do not satisfy.
• "Unbounded means it goes on for ever in all directions." The chapter's own unbounded region is walled on two sides and opens in one direction only. The footnote's phrasing invites the wrong reading; the picture refutes it.
• "An unbounded region means the problem is broken." It means one extra check is needed later. Three of the ten exercise items have unbounded regions and all three have answers.
• "If the region is empty I must have drawn it wrong." Sometimes the demands genuinely cannot be met at once, and the chapter devotes a whole worked example and a whole exercise item to that case. The right response is to prove it arithmetically, not to redraw.
• "The overlap is the union of the shaded parts." It is the intersection. Shading each constraint separately and reading the union is the commonest drawing error, and Fig 12.6 is the figure that exposes it — two shaded pieces and nothing in common.
• "Convex is a technical word I can skip." It is what guarantees the region has corners at all and no dents, which is what makes the next topic's search finite. The chapter asserts it in six words and moves on; an explanation that repeats the assertion without explaining it has said nothing.
• "The non-negative constraints do not need drawing." They are the two lines that confine the region to one quadrant. In Fig 12.5 they are the two walls that stop the unbounded region running away in three more directions.

Questions to check understanding

• Draw the region allowed by a stated system of linear inequalities, and mark its corners
• Decide, for a given point and a given system, whether it is feasible, and if not, name every constraint it breaks
• Say whether a drawn region is bounded, and justify the answer by the chapter's own test
• Given a system, decide without drawing whether its region can be empty, and prove it
• Say which lines in a drawing should be solid and which dashed, given the inequalities they came from
• Produce a point that is feasible but not a corner, for a stated system
• Sort a set of stated systems by whether their regions are bounded, unbounded or empty — the form the whole of Exercise 12.1 supports once it is sorted

Examples worth working on the board

Values marked verified are worked out here from the chapter's own printed data; neither answers file was opened, and this chapter prints no answers to its exercises.

• Fig 12.1 (§12.2.2, Part II p. 397). Read off a three-hundred-dot-per-inch the printed page. The two constraint lines are labelled at the foot of the figure, one of them the simplified investment condition and the other the shelf condition. Both are drawn solid, with arrowheads at both ends, and the shaded region is a single flat tone. The region is a quadrilateral with the origin at one corner, a corner on the horizontal axis at twenty, an interior corner at ten and fifty, and a corner on the vertical axis at sixty; the origin corner is lettered and so are the other three. Note the drawing convention: the axis numeral twenty is omitted from the horizontal scale, and the letter naming that corner is printed in its place. A redraw should print both.
• The three sample feasible points, and what they have in common (§12.2.2, Part II p. 397). The chapter offers the pair ten and fifty, then the pair zero and sixty, then the pair twenty and zero. Verified against the four constraints: all three satisfy every constraint, so all three are feasible — and all three are corner points of the region, with at least two constraints tight at each. The chapter gives no interior example at all. Say so and supply one — see the misconception below — because a student who meets only these three will quietly conclude that feasible solutions live at corners.
• The infeasible sample (§12.2.2, Part II p. 397). The chapter offers the pair twenty-five and forty. Verified: the investment condition demands at most a hundred and this pair gives a hundred and sixty-five; the shelf condition demands at most sixty and this pair gives sixty-five. It fails both, which makes it a weak witness.
• An interior point, supplied by the explanation (not in the book). Verified: the pair five and twenty gives twenty-five plus twenty, that is forty-five, against a limit of a hundred, and a total of twenty-five pieces against a limit of sixty. Both are slack, so the point is strictly inside. Use it as the first feasible example, before any of the chapter's three.
• The shading step the chapter does not teach (§12.2.2, Part II p. 397). The section opens by referring the whole business of graphing a system of inequalities back to Class XI. Verified across the extracted text of all twelve folios: the words test point and origin do not occur anywhere in the chapter, and no rule is given for choosing a side.
• Fig 12.5 and the one dashed line (Example 4, Part II p. 402). Read off a three-hundred-dot-per-inch the printed page. The three constraint boundaries are solid. The fourth line, which is not a constraint but a test line drawn later in the argument, is dashed — and it is the only dashed line in the chapter. The figure also carries two shading tones: the whole unbounded feasible region in a pale tone, and the part of it lying on the far side of the dashed line in a darker one. This topic needs the figure only for the solid-against-dashed contrast and for the unbounded shape; the argument it belongs to is m02-t03's.
• Why the dashed line is dashed (not in the book). Verified: every constraint in this chapter is written with an inequality that permits equality, so every constraint's own boundary satisfies it and is drawn solid. The test line comes from a strict inequality, so its boundary does not satisfy it, and it is drawn dashed. The chapter never says this. The words solid, dashed and dotted occur nowhere in the chapter's text, and the distinction is carried entirely by the drawing. It is the single most important thing in this topic that a text-only reading cannot recover.
• The shape of the unbounded region (Example 4, Fig 12.5, Part II p. 402). Verified from the printed constraints: the region is walled on the left by the condition that the first variable is not negative and below by the condition on the second, and it is capped above by one of the constraints; it runs away only towards the right. So it is unbounded and yet plainly does not extend for ever in every direction. Keep this figure in view while section 7 discusses the footnote.
• Fig 12.6 and Example 5 (Part II p. 403). Read off a three-hundred-dot-per-inch the printed page. The problem asks for a minimum subject to a lower limit on the sum of the two variables and an upper limit on a weighted sum of them. Both boundary lines are drawn solid. There are two shaded regions in two different tones and they do not touch: a small triangle at the origin, and a band on the far side of the other line. Verified: on the triangle allowed by the upper limit, the largest the plain sum of the two variables can be is five, reached at the corner on the horizontal axis; the other constraint demands at least eight. Five is less than eight, so nothing can satisfy both, and the region is empty. The chapter asks the student why and does not answer; supply this argument.
• A drawing note on Fig 12.6 (Part II p. 403). The band drawn for the second constraint is finite on the page, though the region it stands for runs away for ever. That is a space-saving convention and not a claim. A redraw should either carry an arrow showing the region continues or say so in the caption.
• The bounded-and-unbounded footnote (Part II p. 398). The chapter's test is whether the region fits inside some circle. It then adds a sentence characterising the unbounded case which, read with any meaning every, is false of the chapter's own only unbounded figure — see Notes. Give the circle test and stop.
• The convexity remark (Remarks, item (i), Part II p. 403). The chapter asserts that a feasible region is always convex. Verified across the extracted text of all twelve folios: the word convex occurs exactly twice in the chapter — here, and inside a bracket in Theorem 1 on Part II p. 398 — and it is never defined and never proved. Say what convex means, say the chapter asserts it, and say the chapter does not establish it.
• Exercise 12.1, sorted by the shape of the region (Part II pp. 403–404). Verified, region shape only, by working added here of all ten items: six have a bounded region — items one, two, three, five, seven and eight; three have an unbounded region — items four, six and nine; and one has no region at all — item ten, where one constraint puts the second variable strictly above the first and the other puts it at or below, so nothing survives. That sorting is this topic's whole practice set, and it is worth showing, because the chapter never sorts its own exercise.

Figures to have open

• A single line across an empty plane for section 1, with the two half planes tinted and the line drawn in a third treatment. Entirely added here; the chapter draws no such diagram.
• A test-point movement for section 2. Not in the book; the chapter delegates this step to Class XI and draws nothing for it.
• A four-stage build ending on Fig 12.1 (Part II p. 397). The final frame is the chapter's own figure; the build is added here. Print the axis numeral the chapter omits, and letter all four corners.
• A three-point verdict panel for section 5 using the chapter's own four constraints, with an interior point of an added choosing added ahead of the chapter's three.
• A solid-against-dashed pair for section 6, redrawn from Fig 12.5 (Part II p. 402). The distinction is the chapter's drawing and the labelling is added here, since the chapter never names either line style.
• A redraw of Fig 12.5's region for section 7 with the circle test failing on it and the single escape direction arrowed. The region is the chapter's; the circle and the arrow are added here.
• A redraw of Fig 12.6 (Part II p. 403) for section 8, keeping the two tones and the gap between them, but carrying an arrow to show the upper region continues beyond the page — which the printed figure does not.

Where this sits in the book

• NCERT Class 12 Mathematics, Chapter 12 "Linear Programming", §12.2.2 Graphical method of solving linear programming problems, opening paragraph and Fig 12.1, Part II p. 397
• The bold-lead definitions of feasible region, feasible solutions and infeasible solution, with the chapter's four sample points, Part II p. 397
• Optimal (feasible) solution, and the two footnotes defining a corner point and the bounded case, Part II p. 398
• Example 4 and Fig 12.5, for the unbounded region and the one dashed line, Part II p. 402
• Example 5 and Fig 12.6, for the empty region, and Remarks item (i) on convexity, Part II p. 403
• Exercise 12.1, all ten items, sorted here by region shape, Part II pp. 403–404