Hello, amazing math people. We're going to look at compound inequalities or how do we combine 2 separate inequalities. We're going to have two different methods. 1 is going to be an intersection and an intersection is an upside down U and it's referred to mathematically as an and statement. So something has to be in one set and in another set. So if I had a set of 123 and I intersected it with the set 246, the things that are in both sets would just be the number two. If I had the set 123 and I intersected it with the set ABC, did those have anything in common? No. So we'd use a set with nothing in it, or we can use a circle with a slash through. These both are different ways of saying that the set is empty or the empty set. For a solution, we also have what's called a union, which is a capital U and our union stands for OR. It's in one set or it's in the other. So if we look at this examples we did a moment ago, 123 union 246, we're going to list everything that was in either of the two sets. Now we don't need to list that 2 twice. It's plenty to just list it one time. So our answer for an OR would be 12346. If we did 123 unioned ABC, we'd get the set of 123 ABC. We can do the same concept with graphs. So if I had X greater than 7, union X less than 9, less than or equal to 9, if we had X greater than 7, union X less than or equal to 9:00, we'd have a number line. And for this first one here, X greater than 7, we'd go to seven on our number line, and we'd want all of the values greater than 7. If we look at the second part here, we would put in a nine and we want less than or equal to. So we'd want a solid dot, and we'd be shading it to the left. Now, if it's a union, it can be in either of the two sets. So when I look down here, is something shaded? Yes, at the number 7, is one or the other shaded? Yeah. How about in between 7:00 and 9:00 at 9 all the way out to Infinity? So the union of this is actually going to be the entire number line negative Infinity to Infinity. Or sometimes we use a capital R with two lines here at the front to represent all real numbers. What if we did this exact same problem, but this time instead of an intersection or instead of a union, we wanted an intersection? Well, if we want an intersection, what we're looking for is we're looking where they're both shaded or where both the systems are true. If we look at our number line, where are they both true? They're true at 7:00, but not including seven because only one is shaded at 7:00. So we're going to use a parenthesis at 7. They're both true in between 7:00 and 9:00, and they're both true at 9:00, but they're not both true past nine. So our answer to the intersection would be ( 7, 9. Whereas for a union, it was the whole real numbers. Now one more topic we want to look at is domains. And if we have a domain of a function, say F of X equal three X / 5 -, 10 X, we want to know what values we can use. Well, our denominator can never ever, ever be 0. So we're going to say this 5 - 10 X can never equal 0. If we solve for X, we're going to get X can never equal 5/10, which is 1/2. So on a number line, we'd have every single value being true except at this 1/2 location. So we're going to put an open dot at 1/2 and shade everywhere else because it's true everywhere except at that single point interval notation. We'd have negative Infinity to 1/2. We'd use a parenthesis because it's not true at 1/2. We'd use a union 1/2 to Infinity because it's anywhere that's shaded, so we have to use that union. You've done great. Have a wonderful day.