 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 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 three 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 three 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.