 Until now you've probably spent a lot of time looking at equations, but let's talk about inequalities. And, in fact, if you think about it, as a general rule, inequalities are more important than equalities. You'd rather pay less than $100 for an item than exactly $100, or you'd hope to get at least 90% on an exam rather than exactly 90%, or you've planned that a project will take no more than two weeks to complete, and you don't want it to take exactly two weeks. It's useful to be able to move back and forth between algebra and geometry. Algebra is all about numbers and formulas and equations. Geometry is about pictures. And, as the saying goes, a picture is worth a thousand words. So it's helpful to draw a picture of an inequality on a number line, and we'll do so as follows. First, we'll identify the boundaries of the inequality. And, since inequalities come in two flavors, depending on whether we include or exclude the endpoint, we'll use a closed circle to indicate a boundary is included, and we'll use an open circle to indicate a boundary is excluded. And then, once we've indicated the boundaries, we'll shade the portion of the number line that corresponds to the inequality. For example, suppose I want to graph the inequality x greater than 5. Now, the inequality describes all numbers greater than 5, so 5 is a boundary. So let's put down our number line. Since the inequality is strict, we're not allowed to actually equal 5, we exclude 5 and represent it using an open circle. Since we want to include all numbers greater than 5, we shade the portion of the number line that includes these greater numbers. So we shade the part of the number line to the right of 5. That's just one problem. Where do we stop? In fact, we can never stop because we want everything that's greater than 5. But we only have a finite amount of paper, so instead of going to the right forever, we'll use an arrow to indicate that we never stop going to the right. Or how about the inequality? x less than or equal to negative 3. So we'll throw down our number line. And since negative 3 is a boundary, and our inequality allows us to be equal to negative 3, we'll use a closed circle. Since our inequality is x less than or equal to negative 3, we want everything less than negative 3, so we shade to the left. And we want to use an arrow to indicate that we never stop shading to the left. We should also be able to go backwards. So we want to be able to take a graph like this and express it as an inequality. So first, we see that negative 2 is a boundary, but it's excluded because it has an open circle. Next, we see that the shaded portion includes everything less than negative 2. So this corresponds to the inequality x is less than negative 2. We can also graph compound inequalities like negative 1 less than x less than or equal to 2. So we'll throw down our number line. The boundary numbers of our inequality are negative 1 and 2. Let's take a look at them separately. Since negative 1 is less than x, we have to exclude negative 1 and use an open circle there. On the other hand, since x is less than or equal to 2, we include 2 and use a closed circle there. Finally, since we want everything between negative 1 less than x less than or equal to 2, we shade the portion between these two points. Because our interval has a definite beginning and an end, we don't use arrows. Remember, arrows only indicate keep on going, and we don't want to keep on going here. We want to begin here and end here. Again, we should be able to go backwards as well. So suppose somebody produces this graph. First, we see the boundaries are at 1 and 4. We can set these numbers down in the same order that they appear on the number line. So we'll put down a 1 and a 4. Since there's a closed circle at 1, we include 1 in our inequality. Now, we don't want to commit ourselves yet as to whether this is a greater than or equal to or a less than or equal to, but we do know there's an equal to. So we'll put in that little underscore. Meanwhile, since there's an open circle at 4, we exclude 4 from our inequality. So 4 is just going to have a greater than or a less than with it. Finally, we note that we have shaded everything between 1 and 4. Our x has to be between 1 and 4. And it's to the right of 1, so it's got to be greater than or equal to 1 and to the left of 4, so it has to be strictly less than 4. How about another compound inequality, like x less than or equal to negative 3 or x greater than or equal to 5? So we'll throw down our number line. Both negative 3 and 5 are boundaries and both are included. So we use closed circles. We want x less than or equal to negative 3, which means we'll shade everything to the left of negative 3. Or we want x greater than or equal to 5, so we should also shade everything to the right of 5. It's important to note the difference between inequalities that involve or and inequalities that involve and. If we want to graph the inequality between x greater than negative 4 and x less than or equal to 1, we can begin by noting that negative 4 is an excluded boundary and 1 is an included boundary, so we indicate this. Now notice that our boundaries have separated the number line into 3 parts. Also remember that our inequality requires that x be greater than negative 4 and also x has to be less than or equal to 1. So this means we need to shade only those portions where both of these statements are true. So let's take a look at our 3 parts. In the leftmost part, x greater than negative 4 is not true, so we must omit this portion. In the central part, x greater than negative 4 is true and x less than or equal to 1 is true, so we include this portion. And in the rightmost part, x less than or equal to 1 is not true, so we omit it. Now, we don't have to include these cross-outs as part of our graph of the inequality, but it's not a bad idea to keep them as a reminder that we've actually checked these intervals and we know for certain that they are not part of the graph of the inequality. Remember, paper is cheap and there's no harm in recording a lot of information. So even though the graph of the inequality is just this central portion, nothing bad happens if we just leave the cross-outs in place. No, these weren't because somebody left a cross-out in place. Not even this one. It's also important to be able to go back from the graph of the inequality. So here we see that 3 and 5 are boundaries. The shaded regions are everything less than 3, but not including 3, or everything greater than 5, including 5. So the graph shows the inequality x less than 3, or x greater than or equal to 5.