Why testing values one by one is not a method, and what replaces it

What to assume they know

• Everyday constraints that fix a range rather than a value — how a purchasing situation produces a strict inequality, and what its letter counts
• Sorting inequalities: numerical or literal, strict or slack, linear or not — the sorting of inequalities, and what makes one linear in a single variable
• The natural numbers, the integers and the real numbers as three different collections, and which sits inside which
• Listing the members versus stating the property they share — writing a collection by stating the condition its members satisfy
• The number systems as a chain of containments, and intervals as pieces of R — intervals as pieces of the real line, which is the form Example 2 ends in
• Substituting a number for a letter in an expression and evaluating it

What they should be able to do

• Substitute a stated value into an inequality and report whether the statement comes out true
• Distinguish a solution from the solution set, and give both for a stated inequality over a stated collection of numbers
• Say what has to be true about an expression before a single failed test is allowed to close off everything past it
• Give the solution set of one inequality over the naturals, the integers and the reals, and account for the three different answers
• Explain why no finite table describes the answer when the letter ranges over the real numbers
• Produce a solution set that is empty, and say what makes it empty
• State the convention this chapter adopts for the rest of its pages

Where it usually goes wrong

• "I checked six values and they all worked, so I have solved it." Six checks certify six numbers. The answer here is a collection, and nothing in a finite column of checks says what happens outside the column.
• "Once it fails, it fails from then on." True for 30x, because that expression only rises. It is not a property of tables, and the chapter never claims it is. Say out loud why the stopping is legitimate here.
• "The solution set belongs to the inequality." It belongs to the inequality together with the collection the letter may range over. The §5.3 table gives seven values and Example 1 gives six for the same statement, and neither is wrong.
• "Then the integer answer is the better one for the packets." Negative packet counts are arithmetic solutions and nonsense as purchases. The universe was fixed by the situation before the algebra started, and Example 1 is exploring the algebra, not restating the shopping.
• "Every inequality has solutions." Exercise 5.1 item 2 has none over the natural numbers.
• "I could list the real-number answer if I were patient enough." Between any two of its members there is another. The list has no first entry and no last.
• "Trial and error is beneath us." It is how the chapter establishes what a solution is, and it is the only step that connects the symbols to the situation. Its defect is completeness, not honesty.

Questions to check understanding

• Test a stated value in a given inequality and report the verdict
• Give the solution set of a linear inequality over the natural numbers, over the integers and over the real numbers, from the same starting statement
• State a solution set in listed form where that is possible and in interval form where it is not
• Produce an inequality whose solution set over the naturals is empty, or recognise one
• Explain in a line or two why substitution alone cannot establish a real-number solution set
• Identify, from a worded situation, which collection the letter should be allowed to range over

Examples worth working on the board

Items marked verified are worked out here from the chapter's stated data. The book's separate answers file, printed elsewhere in this volume, was not read, checked or extracted at any point.

• The statement under test (§5.3, p. 91): 30x < 200, where x counts 1 kg rice packets. Before any working the chapter fixes what x may be — not negative, not a fraction — on the grounds of what it counts.
• The chapter's table (§5.3, p. 91). Substituting x = 0, 1, 2, 3, 4, 5, 6, 7 gives left-hand values 0, 30, 60, 90, 120, 150, 180 and 210, each set against 200. The first seven come out true and the eighth does not. The collection the chapter records is {0, 1, 2, 3, 4, 5, 6}.
• Verified, and this is section 4's whole argument: the table is entitled to stop because 30x is larger for a larger x, so once the left-hand side has passed 200 it never comes back under. The chapter does not say this. Without it the table proves only what it printed — eight facts about eight numbers — and a student who transfers the habit to an expression that rises and falls will close off values that in fact belong.
• Example 1 (§5.3, p. 92) takes the same statement and divides both sides by 30, giving x < 20/3. Over the naturals the answer is {1, 2, 3, 4, 5, 6}; over the integers it runs downward without end, from 6 through 0 and on past −3.
• Verified: 20/3 sits between 6 and 7, so 6 is the largest whole-number solution in every one of these readings. The natural-number answer has six members; the §5.3 table's answer has seven, because counting there began at zero; the integer answer has no smallest member at all. One inequality, three answers — the difference is entirely in what x was allowed to be.
• Example 2 (§5.3, p. 92): 5x − 3 < 3x + 1. The chapter adds 3 to both sides, then subtracts 3x, reaching 2x < 4 and then x < 2. Over the integers the answer is the numbers 1, 0, −1, −2, −3, −4 and onward downward; over the reals it is written x ∈ (−∞, 2).
• Verified, and this is the sharpest reason testing fails: over the reals the answer has no largest member. Take any r under 2; the number (r + 2)/2 lies strictly between r and 2, so it is a solution too and it is larger. There is therefore no last value for a table to stop at, and no first one for it to start from.
• Exercise 5.1 items 1 to 4 (p. 95) drill exactly this pairing. Item 1: 24x < 100, over the naturals and over the integers. Item 2: −12x > 30, over the naturals and over the integers. Item 3: 5x − 3 < 7, over the integers and over the reals. Item 4: 3x + 8 > 2, over the integers and over the reals.
• Verified, item 1: dividing by 24 gives x < 25/6, which is between 4 and 5, so the natural-number answer is {1, 2, 3, 4} and the integer answer adds everything below.
• Verified, item 2, and it is worth attention: dividing by −12 turns the statement round to x < −5/2. Over the integers that is −3 and everything below; over the natural numbers there is nothing at all. A solution set can be empty, and no amount of substituting will ever discover that — the tester simply keeps failing and never learns why.
• Verified, item 3: x < 2, the same answer as Example 2, so the exercise is a deliberate echo. Verified, item 4: 3x > −6, so x > −2.
• The chapter's own verdict on the method (§5.3, p. 91). It calls the table slow and says outright that a systematic technique is needed, then spends the rest of the section building one. The failure motivates the rules, and the rules are the topic of The one rule that differs from equation-solving, and the reason it differs.

Figures to have open

• A vertical test table for 30x < 200: x in the first column, the left-hand value in the second, the verdict in the third, eight rows. Built from the chapter's own working on p. 91. Standard schematic.
• A rising-staircase plot of the left-hand values 0, 30, …, 210 against a horizontal line drawn at 200, so that "it never comes back under" is visible rather than asserted. An added figure; the chapter argues nothing here.
• Three stacked number lines for one statement, marked for naturals, for integers and for reals, showing dots, more dots running off the edge, and continuous shading. Standard schematic; the chapter's own number-line drawings begin at p. 93 and belong to Drawing the answer as a piece of the number line, hollow circle or solid.
• A "no last member" panel: a candidate r marked under 2, and its midpoint with 2 marked to its right, with the construction repeated once. Not in the book.
• No textbook artwork is needed. Pages 91 and 92 carry no figures — checked on both page images.

Where this sits in the book

• NCERT Class XI Mathematics, Chapter 5 Linear Inequalities, §5.3, whose printed heading names the algebraic solution of linear inequalities in one variable together with their graphical representation, p. 91: the restriction on x, the eight-row table, the naming of solutions and of the solution set, and the verdict that a systematic technique is required.
• §5.3, p. 92, for Example 1 and Example 2, and for the convention that the real numbers are assumed from that point unless something else is stated.
• Exercise 5.1 items 1 to 4, p. 95.
• Summary, p. 99, for the compact restatement of what a solution is.