The one rule that differs from equation-solving, and the reason it differs

0:00Solving an equation uses two moves you have used for years. Add the same amount to both sides. Multiply both sides by the same number, so long as it is not nought. Carry those two over to a statement with an order symbol, and one of them survives untouched while the other does not. The usual telling is that you flip the symbol when you multiply by a negative, and that this is a rule you have to remember.

0:30It is not. It is the same rule as before, said out loud for a statement that has a direction. The whole difference comes from one question: what does the move do to the order of the number line itself? So that is what gets measured. Thirteen numbers either side of nought give seventy-eight pairs where one sits below the other. Every move here goes through all seventy-eight. Start with the move that survives.

1:01Take a pair already in order: minus eight below minus seven. Add ten to each side. They become two and three, and two is still the lower one. Add minus ten instead. They become minus eighteen and minus seventeen, and the order has not moved. Neither of those is evidence on its own. Two instances are two instances. Over all seventy-eight pairs, sliding ten right keeps every one in order: seventy-eight kept, none swapped, none brought together. Sliding ten left gives the same three numbers.

1:40That is what adding equally does: it picks the whole line up and puts it down elsewhere, every point moving the same way. Nothing overtakes anything, so nothing about order can change. That move never touches the symbol. Now multiply, and take the easy case first. Multiply that same pair by two. Minus eight goes to minus sixteen, minus seven to minus fourteen. The gap has doubled, from one to two. The order has not moved.

2:16Halving goes the other way: minus four and minus three and a half, closer together, same order. Across all seventy-eight, doubling keeps every pair in order, and so does halving. Seventy-eight kept, none swapped, none brought together. So multiplying by a positive number is a stretch. It pins nought and pulls every other point away from it, or pushes every point towards it. Distances change. Which side of which anything is on does not.

2:50Same three numbers as the slide, from a completely different picture. Multiply both sides by minus one. Three is above two. Minus three is below minus two. They have exchanged places. Try a bigger multiplier. Minus eight is below minus seven; times minus two gives sixteen and fourteen, and now the first is above the second. Those two instances are usually where the explanation stops. They are true, and they are not a reason.

3:26So put minus one through all seventy-eight. None kept, seventy-eight swapped, none brought together. Minus two: the same. Minus a third: the same. Every pair, every time, without one exception. Now you can say what the move is. Multiplying by a negative stretches the line, exactly as before, and then turns it over about nought. Left and right change places for every point at once, so no pair can be spared.

4:00It is worth asking what it takes for a rule like this to exist. Here is a move with no rule: square the number. Through the same seventy-eight pairs, squaring keeps thirty-six in order, swaps thirty-six, and brings six together. There is nothing you could write down. Not keep the symbol, not turn it. It depends on the pair. A rule about the symbol can only be stated when the move does the same thing to every pair.

4:31Of eleven moves measured here, four keep every pair in order, four swap every pair, and three do neither. One more. Turn the line over and then slide it four to the right: it still swaps all seventy-eight, so the sliding part had no say. So far this is about the line. Bring back the statement. Take x is under three, and hand in six candidate symbols: below, above, at most, at least, equal, and not equal.

5:04For a given move, which of the six, written afterwards, agrees with the original at every value tested? Not most. Every one. That is a search, not a lookup. Nothing says which symbol is the opposite of which; it has to be found. Slide right ten: below. Slide left: below. Times two: below. Times a half: below. Times minus one: the only symbol that agrees everywhere is above. Times minus two: above. Times minus a third: above.

5:40So the turning was never assumed. It was the answer to a search that could have come back with anything. Sometimes it comes back with more than one. A statement true wherever you look is agreed with by three of the six. Now do the same to an equation, and watch what does not happen. Take x equals three, and run the same eleven moves and the same six candidates.

6:08Slide either way, times two, times a half: equals. Times minus one, minus two, minus a third: equals. Every move the equation survives at all, it survives with its symbol untouched. That answers the question nobody asks: why did the equation version of the rule get away with saying only that the multiplier is not nought? Because an equation has no direction. It says two things coincide, and coinciding has no left and right.

6:43Turn the line over and two points on top of each other are still on top of each other. Order is the only thing a flip can hurt. The rule was never missing a condition. There was nothing for the condition to be about. An equation is not indestructible, though, and it is worth being exact about what hurts it. Count the eleven moves again. The order statement survives eight. The equation survives nine.

7:14One move separates them, and it is a strange one: exchange the numbers two and four, and leave every other number where it is. That brings no two numbers together. Nothing collides, so the equation comes through untouched, symbol and all. The order statement does not survive it at all. No symbol works, because the move scrambles order without collapsing anything. Which is the cleanest statement of the difference. An equation survives exactly the moves that bring no two distinct numbers together.

7:52An order statement needs more. It needs the move to treat every pair the same way. Squaring brings six pairs together, one for each number and its negative, so the equation fails there too. Multiplying by nought brings all seventy-eight together. Nought deserves its own moment, because it costs more here than it does for an equation. Take a true equation, three equals three, and multiply both sides by nought. You get nought equals nought: true, and useless.

8:28Now take a true order statement, three is below five, and multiply by nought. You get nought is below nought, which is false. The same move that merely wasted your time in one case has taken a true statement and made a false one. Nought does not just lose information here. It destroys the statement. As a rewriting move it is barred in both settings anyway: run the search and no symbol agrees.

8:58Not below, not above, not at most, not at least, not equal, not unequal. The search comes back empty and the answer is a refusal. Returning some default there would be inventing a rule where none exists. Here is why none of this has to be taken on trust. Solve four x plus three is below six x plus seven. Collect the letters on the left and you reach minus two x is below four.

9:31Divide by minus two, turn the symbol, and x is above minus two. Now solve it again, collecting the letters on the right instead: minus four is below two x, and dividing by two gives minus two is below x. Two routes. One used the turn; the other never went near it. Every step of both, scored against the statement they started from, disagrees at nothing. And the two answers are written with opposite symbols and disagree nowhere.

10:04Do it wrong instead — divide by minus two and leave the symbol alone — and you disagree with the truth at two hundred and forty of the two hundred and forty-one values tried. The one value where they agree is minus two, where both say false. One with fractions, because that puts two moves in the same problem. Five minus two x, all over three, is at most x over six minus five.

10:35Multiply everything by six to clear the sixths. Six is positive, so nothing turns, and you get ten minus four x at most x minus thirty. Gather the letters: minus five x at most minus forty. Divide by minus five, turn the symbol, and x is at least eight. Notice where the turn happened. Clearing the fractions was a stretch; only the last step was a flip. The reversal-free route is there too. Go the other way instead, to forty at most five x, and then eight at most x.

11:14Same answer, no turn anywhere, and every step of both routes disagrees with the original at nothing. Eight itself makes the statement true; just below eight does not. And the answer is positive even though a turn was used, so the sign of an answer settles nothing. Now a statement with two symbols in it. Minus five is at most five minus three x over two, which is at most eight.

11:44Multiply through by two, then take five off everything, and you reach minus fifteen at most minus three x at most eleven. Divide by minus three and both symbols turn. Five is at least x is at least minus eleven over three. Written the ordinary way round, x runs from minus eleven over three up to five, both ends included. The ends have changed places, because the expression in the middle turns the line over: it swaps every one of the seventy-eight pairs.

12:21Check it and the numbers say so. Put five into that expression and you get minus five, the lower bound you started with; put minus eleven over three in and you get eight, the upper bound. Put nought in, which sits between the ends, and you get two and a half, which sits between the bounds. Write the answer the other way round instead and it has nothing in it at all, which costs every one of its fifty-three members.

12:52One more thing to get right: where the rule must not be reached for. Minus twelve is below four minus three x over minus five, which is at most two. There is a minus in there, and the temptation is to flip on sight. Look at what that expression actually is. Three x over minus five is minus three fifths of x. Subtracting it adds three fifths of x: the two minus signs cancel and the coefficient is positive.

13:25Put the middle expression through the seventy-eight pairs and it keeps every single one in order. It is not a flip. So the chain solves with no turn anywhere, giving x between minus eighty over three and minus ten over three, the lower end open and the upper end closed. Turn both symbols on sight of the minus and what you are left with is empty, which costs all one hundred and forty of the values that worked.

13:57The second place is the one every worked example here quietly stays away from. Dividing both sides by the letter. Try it on x is below three. Multiply both sides by x and keep the symbol, and it disagrees at a hundred and eighty-one of the values tested. Turn the symbol instead and it disagrees at sixty. Neither is nothing, and the search comes back with no symbol at all.

14:27You do not know which way x points, so you do not know whether the move stretches the line or turns it over. Which is the whole rule in one line. The turn is not about minus signs, and not about negative answers. It is about whether the move turns the line over, and that has to be one fact about the whole line — which you can only have when the multiplier is a number.