 As a general rule, we're usually more interested in inequalities than inequalities. If you've budgeted $20 for dinner, you want to spend no more than $20. Or, if you have three days to finish a project, you want to complete it in at most three days. And if you want to pass the final exam, you want to get at least a 60. We use inequalities to indicate these relationships. There are four different inequality symbols we use. When we write this, we read this as A is strictly less than B. It must be less. Now, sometimes we drop this strictly and just say A is less than B. But you should remember that when you see this, it cannot be equal. It must be less than B. And how you speak influences how you think. You really should read this as A is strictly less than B. Now, a very closely related symbol is this one, which looks like a less than symbol has been joined to an equality symbol, and we read this as A is less than or equal to B. And the important thing here is that we're not committing ourselves to A being strictly less than B. We're allowing for the possibility it could be equal. We could also write this, and we read this as A is strictly greater than B. It must be more. And again, we often read this as A is greater than B. But you should remember that the strictly greater than is the proper way to read this. And finally, if we want to allow for the possibility of equality, we'll merge the greater than symbol with the equal sign and get this symbol. And we read this as A is greater than or equal to B. And again, this means it could be equal. Now, it's worth noting that inequalities have a directionality to them. In particular, it matters which one's on the left and which one's on the right. If A is less than B, we write A less than B. But at the same time, we also have B is greater than A, and so we can write B greater than A. And what this means is that we can reverse the direction of the inequality by reversing the order of the sides. While less than, less than or equal to, greater than, and greater than or equal to correspond to specific terms in mathematics, we rarely use these terms when speaking. And that means it's going to be challenging to translate a sentence in English into a proper sentence in mathematics. While there's no rule or formula that will always work, here's a strategy that will help you get through most cases. First, find the boundary values. Those values that seem important in the inequality. Now, that might seem to be a little vague, but as it turns out, finding the boundary values will generally be straightforward. Another step that's relatively straightforward is, for each boundary value, find two test points. A value that is greater than the boundary, and a value that is less than the boundary. Now for the difficult steps. These are difficult because they don't involve mathematics, but instead they involve how we speak in natural languages. And unlike mathematics, which is always unambiguous and clear, natural languages are vague and require a lot of reading between the lines. Unfortunately, we have to deal with natural languages all the time, so you might as well get used to it. And so here's where those test points come in. We'll use the test points to determine whether the desired values are greater than the boundary values or less than the boundary values. And finally, and most importantly, we have to decide whether the boundary values themselves are included. So let's see how that works. Let's express as an inequality the length of a rectangle is at least 50 feet. We'll use L to represent the length of a rectangle. Now the only number actually mentioned in the problem is 50, so we'll use that as our boundary value. And now we want to find a number greater than 50 and a number less than 50. And a useful rule here is go big or go home. In other words, if we want something that's greater than 50, don't pick a number like 51. Pick a number like 100,000. And we can also take a number less than 50. And here, another useful guideline is that zero is a good test value, provided it's not a boundary value. So a number less than 50 is zero. And so now let's try to determine if we want the numbers greater than the boundary or less than the boundary. So now remember we want the length of the rectangle to be at least 50 feet. So if the length of the rectangle is greater than 50 feet, say for example 100,000 feet, then this is at least 50 feet. On the other hand, if the length of the rectangle is less than 50 feet, for example zero feet, this is not at least 50 feet. And an important thing to remember, always test the boundary values too. If the length of the rectangle is exactly 50 feet, then this meets the requirement that the length is at least 50 feet. And so what that says is that we want our length to be greater than 50 feet or equal to 50 feet, and so we'll write L greater than or equal to 50. Or let's express as an inequality, a test should be completed in under 60 minutes. Let T be the time it takes to complete the test. So again the only thing that could possibly be a boundary value here is 60. So if the time is greater than 60 minutes, say 10,000 minutes, well this is not under 60 minutes. On the other hand, if the time is less than 60 minutes, let's say zero minutes, this is under 60 minutes. And finally, don't forget to test the boundary values if the time is exactly 60 minutes. This is not under 60 minutes. Since we want that the test should be completed in under 60 minutes, and so the values we want are less than but not equal to 60. And so we want T strictly less than 60.