 Now, sometimes we're subject to several constraints. This produces a compound inequality with AND if all the constraints must be met, or OR if only one is required. For example, let's translate into an inequality. Your age must be at least 18 but younger than 26. So the boundary values are 18 and 26. So the constraint about 18 is that your age must be at least 18. Now if you are more than 18, your age is at least 18. If you are less than 18, your age is not at least 18. And remember, always check the boundary values. So if you are exactly 18, then your age is at least 18. So we want your age to be greater than 18 or equal to it. And so the first inequality is x greater than or equal to 18. How about the other inequality? Well, our boundary value is 26 and the requirement is that you're younger than 26. If you are more than 26, you are not younger than 26. On the other side, if you are less than 26, then you are younger than 26. And don't forget to check the boundaries. If you are exactly 26, then you are not younger than 26. So we want the age to be less than but not equal to 26. So the second inequality is x strictly less than 26. And if we read our requirement carefully, we have to have both these inequalities true. So this is x greater than or equal to 18 and x less than 26. So far we've dealt with simple expressions, single variables, but there's no reason we couldn't express an inequality as it relates to an algebraic expression. So let's translate into an inequality, 3x plus 12 must be less than 50 or at least 100. So again, the boundary values seem to be 50 and 100. So let's consider each of these boundary values separately. So 3x plus 12 is greater than 50, then, well, it's not less than 50. If 3x plus 12 is less than 50, well, then it is less than 50. And again, remember to check the boundary values. If 3x plus 12 is exactly equal to 50, well, then it is not less than 50. And what this means is that we want 3x plus 12 to be strictly less than 50, strict because we're not allowing equality. The other boundary value is 100. So 3x plus 12 is greater than 100, say 5 billion, then that is at least 100. If 3x plus 12 is less than 100, for example, zero, then zero is not at least 100. And again, be sure to check those boundary values. If 3x plus 12 is exactly equal to 100, then it is at least 100. And so what we want for inequality is 3x plus 12 greater than or equal to 100, and we can write it this way. And because the problem said it's either this or that, we'll use the or to join the inequalities. Now it's bad form and we shouldn't do it, but sometimes we can collapse compound inequalities into a single inequality. We can only do this with inequalities that involve and. So let's take a look at one of those. So earlier we saw that the inequality representing the situation that your age must be at least 18 but younger than 26 was x greater than or equal to 18 and x less than 26. To collapse these inequalities, they have to run in the same direction. So we'll make them both less than type inequalities by reversing the first inequality. Now notice that we have x and the same thing in the middle. Now when this happens, we can merge the middle parts into a single thing. And so we might read this 18 less than or equal to x, which is strictly less than 26. And this leads to a general rule, the inequality a less than b and also b less than c can be rewritten as a less than b less than c and similarly for other inequalities. And it's important to remember that in order to do this collapse into a single inequality, the inequalities must run in the same direction. So let's rewrite this as a single inequality. The length must be at least 70 feet but less than 100 feet and we'll use L to represent the length. So the boundary values appear to be 70 and 100. So the length must be at least 70 feet. So if the length is greater than 70, then the length is at least 70 feet. If the length is less than 70, the length is not at least 70 feet. And if the length is exactly 70 feet, then the length is at least 70 feet. And so we want the lengths that are greater than or equal to 70. For the other boundary value, we want to be less than 100 feet. So if the length is greater than 100 feet, the length is not less than 100. If the length is less than 100 feet, then the length, well, is less than 100 feet. And importantly, check the boundaries. If the length is exactly 100 feet, this length is not less than 100 feet. And so we want our length to be strictly less than 100 feet. Finally, since both are required, we need to write this as an AND inequality. So if we want to rewrite as a single inequality, we'll have to rewrite them so that all the inequalities are going in the same direction. So let's rewrite our inequalities so they're all less than type inequalities. So we'll rearrange this first inequality. Now the middle parts are the same. They're both L's, so we can merge the middle, which gives us a single inequality statement.