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

0:00Ten written shapes. Each one is a quantity, a symbol, and something on the other side. Two questions get asked about every one of them. Is the symbol strict or slack? And how many letters are left standing in it, and to what power? Those two questions are independent. Neither answer narrows the other down. Most people meet these ten as a list and try to hold onto it as a ladder: the easy ones, then the harder ones, then the hardest.

0:33It is not a ladder. It is a grid, two rows by three columns, and every one of the six cells has something in it. And the small print nobody reads, the conditions on the coefficients, turns out to be the whole of the rule. All three of those are claims, so we are going to measure them. Start with the cheapest distinction there is: does a letter survive in the statement at all?

1:01Three is less than five. Seven is greater than five. There is nothing to do with either of them. They are true where they stand. Now four more. x is less than five. y is greater than two. x is at least three. y is at most four. Each of those carries one letter, and none of them is true or false yet. Ask one of them to settle itself and it refuses rather than guessing at the letter.

1:30Give the letter values instead. Take thirty-seven numbers, running from minus six to twelve in steps of a half. Those four are satisfied by twenty-two of them, twenty, nineteen, and twenty-one. So the answer to the second kind is a collection of numbers, not a verdict. Hold onto that; it is about to be misused. The first of the two questions, and it is about one symbol out of four.

2:02Less than. Greater than. At most. At least. The difference between them is not in how the sign looks, so do not try to read it off. Ask each one what it does when the two sides are equal. Less than says no. Greater than says no. At most says yes. At least says yes. Two shut the boundary out and two let it in, and that is the whole of strict against slack.

2:28Watch it happen. x less than three is satisfied by eighteen of our thirty-seven numbers. x at most three, by nineteen. They differ by exactly one number, and that number is three: the boundary itself. One point. That is the entire difference between a strict symbol and a slack one. Which brings us straight to the commonest wrong reading of all this: that strict means it admits less. Test it on the four from a moment ago. Twenty-two, twenty, nineteen, twenty-one. The first two are strict; the last two are slack.

3:09Sort them by size and they interleave. Nineteen, twenty, twenty-one, twenty-two, going slack, strict, slack, strict. Neither kind is on top. You could not sort these into two piles by size if you tried. And it reverses whenever you like. x less than five admits twenty-two numbers; x at most three admits nineteen. There the slack one is the smaller collection, and the strict one is the bigger. It gets sharper. x less than three, and x at most two and a half, are satisfied by exactly the same eighteen numbers here.

3:49Now make the steps quarters instead of halves. Seventy-three numbers, over the same stretch. The two come apart at exactly one of them: two and three quarters. So how many is a fact about the number and about which numbers you are allowed. Strict against slack is only ever about one point. The second question now, and it begins somewhere you might not expect. Write out the quadratic shape and count the letters in it. There are four of them: a, b, c and x.

4:25Nothing in the shape itself says which of the four is the variable. That gets decided by what the coefficients are given. Fix a, b and c at ordinary values and one letter is left standing. x, to the second power. Now send a to nought. Same shape as before, not one mark changed, and the second-power term is gone. What is left is first power. Send b to nought as well and there is no letter left in it at all.

4:58The shape never moved. What moved is what survived, and that is the thing worth naming: the variable is not declared, it is whatever is left. One more. Leave c unfixed, and the shape keeps three letters, with c among them, because the side beyond the symbol is part of the statement too. Now the ten shapes, with both questions asked at once. Five of them are strict, and the other five are slack. Those are the two rows.

5:31With ordinary coefficients, four come out linear in one variable, four linear in two, and two quadratic. Three columns. Fill it in. Along the strict row: two, two, one. Along the slack row: two, two, one. Six cells and not one of them empty. Every combination of the two answers actually happens somewhere in the ten. Count the same ten twice over. Five and five going down. Four, four and two going across.

6:04That is what makes it a grid rather than a list, and it is the reason a ladder was never going to work. Stay with the narrow column on the end, the one with two shapes in it. One of those two is slack, and the other is strict. One in each row. That kills a comfortable habit: the feeling that once you have left the straightforward shapes behind, the choice of symbol stops mattering.

6:33It does not. Leaving the linear shapes tells you nothing about which symbol you are holding. And it runs the other way just as hard. Knowing a symbol is strict tells you nothing about how many letters are left or what power they are raised to. Two questions, asked of the same object, and neither one is a hint about the other. Now the part everybody skips. The conditions on the coefficients.

7:03a is not nought. Or, for the two-letter shapes, a and b are not both nought. Is that doing real work, or is it decoration? Hand in five rules and let them compete. The full rule, with every condition. One that forgets the condition on a. One that keeps a and says nothing about b. One that decides by counting the letters. And one that decides by the highest power alone.

7:33Run all five across all ten shapes, with ordinary coefficients in place. The first three agree on every single shape. Ten out of ten, not one disagreement between them. Which is exactly why the small print reads like decoration. At a setting where nothing is nought, dropping it costs you absolutely nothing. So stop using ordinary coefficients. Send a to nought, and run the same five rules again. The full rule now calls four of the shapes nothing at all, and turns the remaining six into one-variable statements.

8:14The rule that forgot the condition on a misfiles every single one of the ten. Not some of them. All ten. And the rule that kept a and said nothing about b? It still agrees with the full rule everywhere. It is not being tested here. So test it. Leave a alone and send b to nought instead. Now the full rule turns the two-letter shapes into one-variable statements, and the second rule breaks on exactly four shapes: those two-letter ones, and nothing else.

8:49Each condition fails at precisely the shapes it was written to protect, and at no others. Across four settings, forgetting the first condition costs twenty misfilings, and forgetting the second costs ten. That is not decoration. The other two rules are worse. They break before any coefficient goes anywhere near nought. Counting the letters gets the two quadratics wrong, because one letter squared is still just one letter. Deciding by the highest power gets the four two-letter shapes wrong, because both letters are first power and a top power of one cannot tell one letter from two.

9:35They break in different places. Two shapes and four shapes, with no overlap: six of the ten between them. Here is a shape that separates all three rules at once. A squared first term, and a plain second letter. Two letters left standing. Counting letters calls it linear in two variables. Highest power calls it quadratic. The full rule calls it neither, and the full rule is right. Two letters is not enough on its own. Both of them have to be first power. That is a condition too, and it is doing work right here.

10:15One honest warning about that two-letter row before we finish. Take a real one. Forty x plus twenty y, at most a hundred and twenty. Two things to buy, and a budget. Match it against all ten shapes by what it works out to, rather than by how it looks, and it fits exactly one of them. And only with its own numbers. Change the forty, or change the hundred and twenty, and the match disappears.

10:46So it has a place on the grid: slack row, two-variable column, one cell, no ambiguity. But now ask which numbers satisfy it, and hand over a value for x alone. All four of the two-letter shapes refuse. One number is not an answer to a two-letter statement. It takes a pair, and what comes back is a region rather than a list. Naming a shape and solving it are two different jobs. This is the naming, and the naming is complete.

11:21Last thing. A single statement can make two demands at the same time. Three is less than five is less than seven. Both halves are true where they stand, and there is no letter in either of them. Now put a letter in the middle. Three at most x, and x less than five. Separately, nineteen of our numbers satisfy the first demand and twenty-two satisfy the second. Together, four. Three, three and a half, four, and four and a half.

11:54And those four are exactly the numbers that meet both demands on their own. The chain is the overlap. It is not a new kind of object. Look at its two ends, though, because they do not have the same standing. Three is in. Five is out. Two less than y, y at most four, and it is the other way about: the lower end is out and the upper end is in.

12:20Which is the grid one more time, in miniature. The symbol on the left and the symbol on the right get answered separately. Two questions. Every time. Never a ladder.