Sorting inequalities: numerical or literal, strict or slack, linear or not

What to assume they know

• Everyday constraints that fix a range rather than a value — what an inequality is, and why some situations produce one instead of an equation
• Reading and writing expressions of the forms ax + b and ax + by, and knowing which letters are the coefficients and which the variables
• What "degree" means for a polynomial expression in one letter
• That a statement with no letter in it is already either true or false
• Why "well-defined" is the whole difference between a set and a heap — what makes a collection well defined
• Listing the members versus stating the property they share — naming a set by the property its members share
• A set with nothing in it, and sets you cannot finish listing — a set with nothing in it

What they should be able to do

• State the four relations that turn two expressions into an inequality
• Sort a given inequality as numerical or literal, and say why only one of the two kinds leaves anything to solve
• Read a double inequality as two simultaneous demands on one letter
• Place any of the chapter's catalogued forms on both classification axes at once
• Explain what happens to ax + b < 0 when a is zero, and hence why the linear definition carries that condition
• Explain why the two-variable definition names conditions on both coefficients
• Identify a quadratic inequality and say which condition keeps it out of the linear family
• Say which of the catalogued families this chapter goes on to solve

Where it usually goes wrong

• "Strict versus slack is about how big the answer is." It is about one point. Over the real numbers x < 3 and x ≤ 3 each contain infinitely many values and differ by exactly one — the number 3 itself.
• "3 < 5 is not a real inequality, there is no letter in it." It is; the book gives such statements their own name. What it lacks is not status but a question.
• "A double inequality is two problems stapled together." It is one statement making two demands at the same time, and both must hold. Its two ends can carry different endpoint status, as 3 ≤ x < 5 does.
• "Anything that is not linear must be slack." The catalogue contains (13), slack and quadratic, and (14), strict and quadratic. Knowing a statement is quadratic tells you nothing about its relation symbol.
• "ax + b < 0 is linear whatever a is." With a = 0 there is no x in it. The condition is not decoration; it is what makes the family well defined.
• "Two variables just means twice as much work." This chapter never solves a two-variable inequality. The forms are named on p. 90 and then dropped.
• "The exponent in (13) is a typing slip." It is printed on the page. The extraction loses it; the image does not.

Questions to check understanding

• Sort a list of given statements as numerical or literal
• Sort a list of given statements as strict or slack
• Say whether a given statement is linear in one variable, linear in two, or neither, and name the condition on the coefficients that decides it
• Say what a stated form becomes when a named coefficient is zero, and whether it is still an inequality in that letter
• Read a double inequality as two separate demands, and state the status of each of its two endpoints
• Given a worded situation, write the model and then place it in the classification

Examples worth working on the board

Items marked verified are worked out here from the chapter's printed data; the book's separate answers file was not consulted.

• Definition 1 (§5.2, p. 90). Two numbers, or two algebraic expressions, put into relation by <, >, ≤ or ≥, make an inequality. State it in an added construction — the printed sentence is a formal definition and must not be carried across.
• The chapter's numerical examples (§5.2, p. 90): 3 < 5 and 7 > 5. Verified: both are true as printed, and neither contains a letter, so there is nothing to find — the statement's job is done the moment it is written.
• The chapter's literal examples (§5.2, p. 90): x < 5; y > 2; x ≥ 3; y ≤ 4. Each is a demand on a letter, and each has a whole range of numbers meeting it.
• The chapter's double inequalities (§5.2, p. 90): 3 < 5 < 7, which is numerical; and 3 ≤ x < 5 and 2 < y ≤ 4, which are literal. Text-layer trap: the extraction of p. 90 flattens both slack signs in these three chains into strict ones. The relations above were read off the page image. Anyone rebuilding these from pages/p90.txt will get the endpoint status wrong in two of the three chains.
• Verified: 3 ≤ x < 5 is two demands at once — x is at least 3, and x is under 5 — so its endpoint status differs at the two ends, closed on the left and open on the right. That asymmetry is what makes double inequalities worth a name of their own.
• The catalogue (§5.2, p. 90), printed as ten numbered forms. Numbers (5)–(8) are ax + b related to 0 by <, >, ≤, ≥ in that order. Numbers (9)–(12) are ax + by related to c by the same four in the same order. Numbers (13) and (14) carry a squared term: ax² + bx + c ≤ 0 and ax² + bx + c > 0. Second text-layer trap: the extraction of (13) drops the exponent and reads it as if the letter appeared twice unsquared; the printed form was read off the image.
• The book's own sorting of that catalogue (§5.2, p. 90). Strict: (5), (6), (9), (10) and (14). Slack: (7), (8), (11), (12) and (13). Linear in one variable: (5) through (8), on the condition a ≠ 0. Linear in two variables: (9) through (12), on the condition that neither a nor b is zero. Not linear: (13) and (14), which the book identifies as quadratic in one letter when a ≠ 0.
• Verified by counting the book's own lists: ten forms in all, five strict and five slack; four one-variable linear, four two-variable linear, and two quadratic. The two counts partition the same ten forms in two different ways, which is the evidence that the axes are independent.
• Verified, and this is the argument of section 7: put a = 0 into ax + b < 0 and what remains is b < 0. No letter survives, so the statement is not about x — it is true for every real x if b happens to be negative, and true for no x otherwise. A statement whose solution set is either the whole line or nothing at all is not solvable in any useful sense, and excluding it is exactly what a ≠ 0 does.
• Verified, section 8: put b = 0 into ax + by < c with a ≠ 0 and what remains is ax < c, which is a one-variable statement wearing two-variable clothes. Put a = 0 instead and by < c is the same thing in y. So a definition that named only one of the two coefficients would let one-variable statements into the two-variable family.
• Verified, section 9: put a = 0 into ax² + bx + c > 0 and it becomes bx + c > 0, which lands in the linear family — but only if the coefficient on x is itself non-zero. Set both to zero and the statement degenerates exactly as section 7 describes for ax + b < 0: no letter survives, and what is left is true for every real number or for none. So the book's parenthetical condition on the squared coefficient is what separates (13) and (14) from (5) through (8), and the non-zero condition the book attaches to the linear forms is what keeps the collapsed statement inside them.
• The two-variable model from the previous topic, 40x + 20y ≤ 120 (§5.2, p. 89), sits in the catalogue at form (11). Use it to show that the abstract forms are the models of the opening page in general dress.

Figures to have open

• The two-way classification grid described in the visual treatment, carrying all ten of the chapter's numbered forms. This is an added construction: the book prints the catalogue as a list and the sorting as prose, and never draws it. It is the figure the topic cannot be taught without.
• A pair of annotated forms showing a coefficient being set to zero and the statement collapsing — one for the one-variable case, one for the two-variable case. Standard schematic.
• A single number line with one boundary point drawn twice, hollow and filled, to carry the strict-versus-slack axis before the number-line topic develops it. Standard schematic; the chapter's own number-line figures do not appear until p. 93.
• No textbook artwork is needed. Page 90 carries no figure — checked on the page image.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 5 Linear Inequalities, §5.2 Inequalities, p. 90: Definition 1; the numerical, literal and double examples; the catalogue of ten numbered forms; the strict-and-slack sorting; the linear conditions on a and on a and b; and the identification of the last two forms as quadratic.
• §5.2, p. 89, for the two-variable purchase model that instantiates form (11).
• Forward pointer inside this chapter: the family this chapter actually solves, linear in one variable, is taken up at §5.3, pp. 91–94.