 So we could use the number line as a geometric representation of the algebraic idea of a number. How about for the geometric representation of the algebraic idea of an inequality? So let's pose this as a problem. Let's graph 5 and 8 on the number line, and then find the algebraic relationship between them, and the geometric relationship between them. So we'll set down our origin, and then draw a horizontal line through it to represent our number line. And now let's graph our two numbers. Since both 5 and 8 are positive, both of them are to the right of the origin. But remember the number corresponds to the distance from the origin, and 8 is going to be farther to the right than 5. So we want to say something about the relationship between 5 and 8, both algebraically and geometrically. So algebra is all about formulas and symbols and equations and variables, and the inequality we might write here is that 5 is less than 8. Geometry, on the other hand, is about pictures. And so if we look at our picture, we see that we can say that 5 is to the left of 8. And if we look at this, this suggests the following idea. If A is to the left of B on the number line, that's a geometric idea, then A is less than B, which is our algebraic equivalent. Alternatively, we can look at it from the other direction. If B is to the right of A on the number line, then B is greater than A. And what this means is this reduces this problem of the algebraic inequality to the geometric position on the right or left. For example, let's graph 8, minus 2, and minus 5 from the number line, then determine the correct inequality relations among them. So let's put down our origin and number line. So 8 corresponds to the point 8 units to the right of the origin. So we'll draw it. Minus 2 corresponds to the point that's 2 units to the left of the origin. And finally, minus 5 corresponds to the point 5 units to the left of the origin. Now since 5 is greater than 2, this point should be farther away than the point that's 2 units to the left of the origin. So it's helpful to remember what the connection is between the algebra and the geometry. If A is to the left of B, then A is less than B. If B is to the right of A, then B is greater than A. So on our number line, we see that minus 5 is to the left of negative 2. So negative 5 is less than negative 2. Negative 5 is also to the left of 8, so we have negative 5 less than 8. How about negative 2? So we see that negative 2 is to the right of negative 5, and so that means that negative 2 is greater than negative 5. On the other hand, negative 2 is to the left of 8, and so we write, negative 2 is less than 8. Finally, we see that 8 is to the right of negative 5, so we write, 8 is greater than minus 5, and 8 is also to the right of negative 2, and so 8 is greater than minus 2.