 So it is almost always easier to draw the number line. And remember, on a number line, the value is the distance from 0, and the sign indicates whether you are to the right or to the left. So let's try to compare the absolute value of negative 5 and negative 5 itself. So the hard way, find the absolute value of negative 5, which is 5, and negative 5 is less than 5, so negative 5 is less than absolute value of negative 5. But let's try to do things the easy way. So we can plot negative 5 on the number line, we can plot the absolute value of negative 5 on the number line, so remember, absolute value only changes the sign, never the magnitude, so the absolute value of negative 5 will be in the same place as 5, and since negative 5 is to the left of the absolute value of negative 5, then negative 5 is less than the absolute value of negative 5. So the reason that this is useful is that we don't even have to know what these numbers are. Suppose x is negative. Well, let's compare the absolute value of x and x itself. So again, we could do this the hard way or the easy way. Actually, why would you want to do it the hard way? Let's do it the easy way. x is negative, so it's to the left of 0. So we'll plot it. Remember that the absolute value of x is the same distance from 0 as x is, but always on the right side of 0. So that means absolute value of x is the same distance from 0, but always on the right. So we'll plot that. And since x is to the left of the absolute value of x, we know that x is less than the absolute value of x. Suppose x is negative. Well, let's compare the absolute value of x and negative x. So we'll do this the easy way. We'll draw the number line. Since we're told that x is negative, we can draw it. We know it's to the left of 0. Now, if we want to say something about negative x, we know that negative x will be positive, so it's going to be to the right of 0. And since the only thing that's changed is a sign, the only thing that's changed is whether we're on the right or the left. We've got to be the same distance away from 0 as x was. Meanwhile, the absolute value of x is the same distance from 0 as x, but always to the right. And the thing to see here is that minus x and the absolute value of x are at the same place, and so they must be equal. Or what if we have x, y, and z, and we're actually showing them on the number line? Let's put a bunch of expressions in order from least to greatest. Now, since we want to do this in the hardest way possible, we'll skip trying to graph the expressions on the number line. Wait, actually the problem doesn't say to do things the hardest way possible. Maybe we don't want to do things the hardest way possible. Since we want to do this the easiest way possible, let's graph these expressions on the number line. We've already plotted or have been given x, y, and z. The absolute value of x is the same distance from 0 as x is, but always to the right. So that's going to be about here. The absolute value of y is the same distance from 0 as y is, but on the right. But since y itself is already on the right, then absolute value of y and y are going to be in the same place. The absolute value of z is the same distance from 0 as z, but to the right. And again, since z is already on the right, it's going to be in the same place as z. And the only complicated one is this, what the absolute value of x minus y. So remember, this is the distance between x and y. That's going to be this amount here. So remember, the value of a number when we plotted on the number line is the distance from 0. So we want something that's this far from 0. And so the absolute value of x minus y is going to be right about here. And remember on the number line, as we go from left to right, our values increase. So reading from left to right, so the left most value is x, that's the least, which is going to be less than the absolute value of y, which is equal to y, which is less than the absolute value of x, which is less than the absolute value of z, which is equal to z. And the right most expression, the largest of these all, is this absolute value of x minus y.