Drawing the answer as a piece of the number line, hollow circle or solid

0:00Solve a statement about a letter and the answer is not a number. It is a stretch of the line. Writing it out is not on offer, so you draw it instead, and the drawing has to carry everything the answer says. A drawing here is three things and nothing else. A place, which is where the answer stops. A side, which is the way the heavy stroke runs from that place.

0:26And a mark on the place: a small circle, drawn either hollow or filled in. All of this is about that circle. What it is for, and why the shading cannot do its job for it. Everything gets scored the same way, against a hundred and one values spread evenly from minus twelve to thirteen. A count over those hundred and one is a fact about where we looked, and it will be said that way every time.

0:58Take seven x plus three is below five x plus nine. First question. Where does the answer stop? At the place where the two sides are equal, and across the whole range there is exactly one such place: three. Second question. Which way does the stroke run? Ask the statement about a value a quarter either side. At two and three quarters it says yes. At three and a quarter it says no. So the stroke runs left.

1:30Third question. Is the endpoint itself in? Ask about three. The left side comes to twenty-four. So does the right. Twenty-four is not below twenty-four, so three is out, and the circle is hollow. Nothing in that read a symbol off the page. Three questions, three answers, and the drawing is finished. Scored against the statement it came from, that drawing agrees at every one of the hundred and one values.

2:01And the drawing is not a fact about how the statement was written. Three x minus two is below two x plus one takes different values on both sides, disagrees with the first statement about nothing, and draws identically. Now ask what each of the three pieces is actually carrying. With the place fixed at three there are four drawings you could make: stroke left or right, circle hollow or filled.

2:32Score all four against the statement. The disagreements come out nought, one, a hundred, and a hundred and one. One of the four is right, which is no surprise. What is worth noticing is the spacing. Turn the stroke round and you are wrong about a hundred values. Change the circle and you are wrong about one. One value out of a hundred and one. That is the whole of what the mark carries.

3:01And it is not a value scattered somewhere in the middle. The one value the mark decides is the endpoint itself. So why not leave the circle off altogether and let the shading say it? Here is that drawing: a place at three, a stroke running left, and no mark. Ask it about any value and it answers, with exactly one exception. About three it refuses, because there is nothing there to read.

3:32About the other hundred it agrees with everything below three. And about the same hundred it agrees just as well with everything up to and including three. Those are two different answers. They disagree at exactly one value, and it is the one value the picture will not discuss. The shading is not being careless. It has nowhere to put the information. The circle is not decoration added to a drawing that was already complete. It is the missing half of it.

4:08You might think the shading could carry it anyway, by stopping in the right spot. Run right up to three and stop just short. Just short of what, though? Take any solution and go halfway from it to three. Start at nought. Halfway to three is three halves, which is still a solution, and it is larger. Do it again and you get nine quarters. Again, twenty-one eighths. Then forty-five sixteenths, ninety-three thirty-seconds, a hundred and eighty-nine sixty-fourths.

4:42Six steps, every one a solution, every one larger than the last, and not one of them reaching three. There is no last solution for the shading to stop at, so there is nowhere for it to stop. The construction is not a trick that works wherever you point it, either. Aim it at five instead of three and the first step, five halves, is still a solution, but the second one is not, and the climb refuses rather than quietly stopping.

5:14And that is a fact about which numbers the letter is allowed to take, not about the statement. Over the whole numbers the same statement does have a last solution: two. The halfway step from two is five halves, which is not a whole number, so the construction stops dead. Here is a second statement, worked exactly the same way. Three x minus four, all over two, is at least a quarter of x plus one, take away one.

5:47The two sides meet at one place: one. At three quarters the statement says no, and at one and a quarter it says yes, so this time the stroke runs right. At one itself both sides come to minus a half. Equal — and this statement admits equal. So the circle is filled in. Same line, same three questions, and a drawing that differs from the first one in all three of its pieces.

6:18Which is exactly why those two cannot teach you what the circle means. Two drawings that differ in three ways at once tell you nothing about which difference is doing what. So take a pair built to differ in one way only. Five x minus three is at least three x minus five. And three times one minus x is below two times x plus four. Both meet at minus one. Both refuse below it and admit above it, so both strokes run right.

6:52The only thing left to differ is the circle: filled for the first, hollow for the second. Count the pieces of the drawing that differ and the answer is one. And the two statements themselves disagree about exactly one value out of the hundred and one. Minus one, the place they share. The same information gets written down as well as drawn, using brackets. The first answer is everything below three, with a round bracket at three. The second is everything from one upward, with a square bracket at one.

7:31Score each written form against its own drawing and the disagreements come out nought, all four times. Now turn one bracket round. Round to square, or square to round, and score again. The cost is one value. Every time, for every one of them. One value is what the circle is worth, one value is what the bracket is worth, and it is the same value both times. A round bracket is the hollow circle written down. A square bracket is the filled one.

8:06They are not two styles for the same thing. They are two ways of writing the one bit of information the shading cannot hold. And it reads backwards just as well. Handed a drawing, hollow becomes a round bracket and filled becomes a square one, with the place written between them. One more single statement, because an endpoint is under no obligation to be tidy. Half of x is at least a third of five x minus two, less a fifth of seven x minus three.

8:42The two sides meet at minus two sevenths, which sits on no tick at all. Ask about that endpoint and the statement admits it, so the circle is filled. Ask a quarter either side and only the upper one comes back yes, so the stroke runs right. Three questions again, and the same three answers back. Where the endpoint happens to land is not the circle's problem. Now put two constraints on the same letter at the same time.

9:15Three x minus seven is below five plus x. And eleven minus five x is at most one. Each of those is a statement in its own right, and each gets a line of its own. The first admits seventy-two of the hundred and one values. The second admits forty-five. Held together they admit sixteen. Which is why shading one line and stopping is not an answer to the question that was asked.

9:45Draw the first, draw the second under it, and then draw a third line: the stretch where both strokes cover the same ground. Three lines for two constraints, and the answer is the bottom one. The answer runs from two up to six, and its two ends are marked differently. Here is where each mark came from. Put two into the first constraint. The left side comes to minus one and the right to seven — true, with room to spare.

10:18Put two into the second. Both sides come to one. Equal, and this constraint admits equal, so two is in. Now put six into the first. Both sides come to eleven. Equal, and this constraint refuses equal, so six is out. Put six into the second and it is nowhere near tight: minus nineteen against one. So exactly one constraint is tight at each end, and it is a different constraint at each end.

10:50The one that is tight is the one that decides, and what it decides is whether that number is in the answer. Written down it is a square bracket at two and a round bracket at six. Either other pairing of those ends costs a value. Four more pairs, to see what shapes an overlap comes in. The first gives everything strictly between minus five and five, both ends hollow.

11:19The second gives everything strictly between minus one and seven, again both hollow. The fourth gives everything from minus seven to eleven, and this time both ends are filled. The third is the interesting one. One constraint gives everything above minus five, the other everything above five. The second swallows the first. Every value the second admits, the first admits too, and not the other way round. So the overlap is the second constraint on its own, and it has no right-hand end to mark at all.

11:57Meanwhile the first constraint admits forty values that the answer does not. An overlap is not always a segment between two ends. Two last things about what is on the page. The arrowhead does not mean the stroke stops there. It means the drawing ran out of room. Ask the first drawing about minus forty, or about minus four thousand, and it still says yes. And all of this happens on one line, because the answer to a statement in one letter is a set of numbers, and numbers live on a line.

12:36So: a place, a side, and a mark. The place and the side you could have worked out from the shading alone. The mark you could not. It is worth one value out of a hundred and one, and it is the only thing on the drawing that separates below three from up to and including three.