now in the last section, we started talking about compound inequalities, and we don't when they were or statements. Well, there's another type of compound inequality that involved and statements. So that's what we're going to talk about in this section. So, just like in the last section, we're going to start out with the sentence. All real numbers greater than negative, too, and less than four. And what we're gonna do is try and write a compound inequality to represent it. Now it's sometimes might be helpful to think about what this would look like on the number line, so let's think about the numbers that we're talking about. The real numbers have to be greater than negative, too. And that's the key word. They have to be less than for Well, think about what the word and means. That means that these values or these real numbers, have to fit both d scenarios. So we have to think of numbers that are bigger than negative, too. But at the same time, they have to be less than four. So what numbers might we be talking about with zero fit the case? It would because zero was bigger than negative, too. And it's less than four. Same thing one would work to woodwork. Three will work all the way up till four now with four work. No, because it says strictly less than so, we would have an open circle at four. But it's including all these values that are less than four down until you get to negative two again, we're gonna have an open circle because we're saying that we have to be greater than negative to not equal to it. So let's see what we have for shading here. It looks like we're shading all these values in between these two end points if you want to think of it that way because all of these values are greater than negative, too, which is why we've shaded to the right of negative, too. But they're also at the same time, less than four. Which is why we've shaded to the left of four. Okay, so now the question is, well, how would we write and inequality to represent this? Well, let's start by looking at the first half. How would we write it? Inequality to represent that were greater than negative too. Well, that would mean we would put X is greater than not equal to negative, too. Well, then, they ever part is that we have to be less than four. Well, the inequality to represent that would be X is less than four. And we're gonna put our keyword in here. And so this would be our inequality X is greater than negative too, and X is less than four. Now, with your statements, you always have to have the word or in between, and they're going to be two separate inequalities. Now, the one thing we can do it and statement is we can actually combine them into one. Take a look at what's happening when you're number line. Here. All these values that are shaded here are all different values of ax and notice how all these values of acts are in between negative two and four. So we're gonna somehow right one inequality to represent this. So here's what I'm gonna dio. I'm going to take this first. Inequality X is greater than negative, too, and I'm gonna flip it around. Never words. I'm gonna write the negative to first and the ex second, but notice how the arrows pointing to the negative, too. So I'm gonna have it pointing to the negative Tua's Well, now our second inequalities. All set. It says X is less than four. So here's what we do to join them. We're gonna bring down our first value. Negative, too. No, that's it's less than next. We're gonna put our less than and then X and then our second equality says the same value for X is also less than four. So we're then going to say it's less than four. So this is also the same inequality just written in a different way, where we don't use the word and but know that these two things mean the same thing. So it's a lot of times you'll be given the compound inequality as this one right here and know that we can break it up. It will always be in an statement. So now let's say we had the inequality. Negative three is less than or equal to X, which is less than five. So let's say if you wanted to go ahead and graft us so what? We would dio Sorry, curse. I got away from me here. So what we can do first is break these off into two separate inequalities, so start with the first one. It says negative three is less than or equal to X, and at the very same time X is less than five. So that's what this inequality compound inequality is saying now. In terms of graphing, it might be helpful to flip around our first one, so if you flip it around, it's now going to be X is greater than or equal to negative three. So we have to graph all the values of X that are greater than negative, greater than or equal to negative three and at the same time are less than five. So let's set apart inequality or sorry, our number line. I'm gonna have negative three and five is our critical values. So if we're greater than equal than negative three, we could be negative three. So we'll have a close circle. And if we're greater than that means we would be shaving to the right. But at the same time, we have to be less than five. Meaning would have an open circle it five, and because we're less then we would have to go to the left so know this all are shaded. Area is in between these two values. And again we could try double values to show that we wouldn't shade the outsides because, for instance, if I was to take the number 77 is not less than five. Therefore, it would not make the statement true because it's an and statement. Similarly, let's say I took the value of negative seven. Well, negative seven is not greater than equal to negative three, which is why that number is not in our shaded area. So what? You'll find 99% of the time when you're shaving and statements you're going to shade in between the two values. And when it comes to solving compound inequalities, they're going to be very similar. A lot of times you'll be given the two words or the word and in between. And just like what you're or statements, you'll solve them both separately. Sorry, the screens getting away from me now. The only other difference that you might see is you might be asked to solve compound inequalities. That air just written as one. So let me show you one of those here. So, for example, let's say we were asked to solve negative eight. Sorry. Negative seven is less than or equal to X plus three. And this is less than nine. So what you could dio is you could break them off into two different inequalities or we could solve it all at once. Our goal is to get X all by itself in sat in the middle of these two inequalities. So how would we move that positive three? Well, we would have to subject three from both sides. But when you do it, you have to do it from all three sides of your inequality. That's the key part if you're gonna choose this one because the three ministry that cancels. So now let's start on the left. We have negative seven minus tree. Well, that's negative. 10 we'll bring down are less than or equal to sign in the middle, all we're left with his ex And on the right hand side, we have nine minus tree, which is six. So now we've solved our compound inequality and again, the graph it. The nice thing about thes is notice how your exes in between Negative 10 and six. That's the hint We're gonna shade in between those two values. So we'll put our T values on our number line at negative 10. We're gonna have a close circle because it's less than or equal to X, and that six will have an open circle because it's just strictly lesson. And like I mentioned before, the nice thing about these is you always shade in between. So in the next couple of videos will show you some examples of how you can solve compound inequalities.