Everyday constraints that fix a range rather than a value

What to assume they know

• Forming an expression in one or two letters from a word description, and reading it back as a quantity
• Solving a linear equation in one variable, and what it means for an equation to have exactly one solution
• Whole numbers versus fractions, and the idea that some quantities can only be counted in whole units
• Reading the four order symbols aloud: less than, greater than, and their two "or equal" companions
• Why "well-defined" is the whole difference between a set and a heap — what makes a collection well defined, so that membership has a settled answer
• Listing the members versus stating the property they share — naming a set by the property its members share

What they should be able to do

• Give two situations that cannot be written as equations, and say what blocks the equality in each
• Build the spending expression for a purchase from a unit price and a count, in one letter and in two
• Decide whether a stated budget can be spent exactly, and use that decision to choose between a strict and a slack symbol
• Show that a slack statement carries a strict part and an equality part, and produce an instance of each
• List the whole-number purchases that spend a given budget exactly, from the budget relation alone
• Explain why an inequality's answer is a range rather than a number, without yet solving one
• State what this chapter goes on to solve and what its opening pages promise but never deliver

Where it usually goes wrong

• "≤ is just < being cautious." For Reshma the boundary is hit in four separate purchases; for Ravi it is hit in none. The symbol is a claim about whether the endpoint is reachable, and here the two situations answer that question differently.
• "If ₹200 is available, ₹200 is what gets spent." Goods sold in indivisible units make most totals unreachable. Ravi's best purchase leaves ₹20 behind, and no purchase does better.
• "Both are 'less than something', so both take <." Reshma's takes ≤. The test is not the English wording but whether some allowed purchase reaches the limit.
• "An inequality is a vaguer statement than an equation." It is a different claim, not a weaker one. 30x < 200 says something exact about every value of x; it just happens to say yes to more than one of them.
• "x could be any number here." x counts packets, so it is a whole number, and §5.3 (p. 91) says so before doing any algebra. The permitted universe is part of the model, not an afterthought.
• "The chapter will show me how to solve Reshma's two-letter statement." It will not. Reshma's model is built on p. 89 and never solved anywhere in the printed chapter.

Questions to check understanding

• Turn a stated purchasing or capacity limit into an inequality in one or two letters, and justify the symbol chosen
• Decide, for a given unit price and budget, whether the budget can be spent exactly, and give the remainder when it cannot
• List all whole-number purchases meeting a stated budget exactly
• Say which of a set of given statements are equations and which are not
• Explain in one or two lines why a stated situation cannot be modelled by an equation
• Split a given slack statement into its strict part and its equality part, and give one instance satisfying each

Examples worth working on the board

Items marked verified are worked out here from the chapter's stated data. This chapter prints no answer key on these pages, and the book's separate answers file was not consulted.

• The two opening statements (§5.1, p. 89). Every student in the class stands under 160 cm. A classroom accommodates no more than 60 tables and chairs counted together. Both fix a ceiling and leave everything below it open; neither names a height or a count.
• Ravi's purchase (§5.2, p. 89). ₹200 in hand; rice sold only in packets of 1 kg; ₹30 per packet; x counts the packets; the outlay is ₹30x; the model the book writes is 30x < 200. The book asks the reader why the whole amount cannot go, and leaves the answer out.
• Verified, and this is the argument: 200 ÷ 30 is 20/3, which lies between 6 and 7 and is not a whole number. Six packets cost ₹180 and leave ₹20; seven cost ₹210, which he does not have. So there is no whole-number purchase at all that lands on ₹200, and ₹20 is the smallest change he can be left with. Equality is not merely improbable here — it is impossible, which is exactly what earns the strict symbol.
• Reshma's purchase (§5.2, pp. 89–90). ₹120 in hand; one register ₹40; one pen ₹20; x counts registers and y counts pens; the outlay is ₹(40x + 20y); the model the book writes is 40x + 20y ≤ 120, because her spending may go all the way up to the budget.
• Verified: dividing the equality case by 20 gives 2x + y = 6, and the whole-number purchases with neither count negative are (x, y) = (0, 6), (1, 4), (2, 2) and (3, 0) — four ways to spend ₹120 to the rupee. Show those four as a short table. They are the evidence that her ≤ is doing work: the boundary is reached, four times over.
• Verified contrast to make the point land: 30x = 200 has no whole-number solution at all, while 40x + 20y = 120 has four. One boundary is attainable and one is not, and that single difference is what separates the two symbols.
• The book's own decomposition (§5.2, p. 90). It splits the slack statement into two labelled statements — the strict one, 40x + 20y < 120, and the equation, 40x + 20y = 120 — and points out that only the second is an equation.
• Verified instances for that split: (x, y) = (1, 2) spends ₹80, so it satisfies the strict half; (x, y) = (2, 2) spends ₹120, so it satisfies the equality half. Both are legal purchases and only one of them empties the purse.
• Definition 1 (§5.2, p. 90) is the chapter's formal statement of what an inequality is: two numbers, or two algebraic expressions, set in a relation by one of the four order symbols. State it in the wording used here; the detailed sorting that follows it belongs to Sorting inequalities: numerical or literal, strict or slack, linear or not.

Figures to have open

• A budget bar for Ravi: total length ₹200, six ₹30 blocks laid along it, the ₹20 tail shaded differently, and a seventh block shown overhanging the end. Standard schematic; the chapter prints no figure in §5.1 or §5.2.
• A four-row table of Reshma's exact-spend purchases, (0, 6), (1, 4), (2, 2), (3, 0), each with its ₹120 total. Standard schematic, built from the chapter's data. Keep it a table: do not plot it as a shaded area in the plane, because that treatment was removed from this book.
• A two-column panel setting Ravi's statement beside Reshma's, with one row per question: what is being counted, what the ceiling is, can the ceiling be hit. Standard schematic.
• No photograph or textbook artwork is needed. Pages 89 and 90 carry no figures at all — checked on both page images.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 5, printed title Linear Inequalities; §5.1 Introduction, p. 89, for the two opening ceiling statements and the four order symbols.
• §5.2 Inequalities, pp. 89–90, for Ravi's purchase and its model, Reshma's purchase and its model, the split of the slack statement into two labelled statements, and Definition 1.
• Forward pointer inside this chapter: the promised universe for x, and the first solving of Ravi's statement, are at §5.3, p. 91.