 This video is going to talk about linear inequalities. So a linear inequality looks like ax plus b is greater than c. Again, it looks like a lot of variables, but we really just have this one variable of x and a, b, and c are going to be real numbers. And again, we can't let a be equal to zero like in equations because if it were, then we wouldn't have a linear equation. We just have a bunch of numbers. So we're looking at graphing, and we want to graph x is less than or equal to one. Well when we graph, we can use, some of you might have learned using closed and open circles, but we're also going to talk about interval notation, which deals with parentheses and brackets. And we might as well just talk about that now. If you have parentheses, it means that your endpoint is not included. And if you have brackets, it means your endpoint is included. So I'm looking at this, and it can be equal to one. So when I come here to one, it's going to be a bracket. And it's going to be less than one, so it's going to be, I'm going to open my bracket to the left because that's where my numbers are less than one. And I would put my bracket in here, and then I'm just going to put my arrow here. And that means that it goes on and on and on and on and on. Now, if I want to write the interval notation, again, we have to think about the endpoints. And it also, we've talked about the endpoints, but come down here and it says the smallest interval and then the largest interval. That's how we write them. Well, down here, this is negative infinity at the end of that arrow because it just goes forever. That's our smallest value. The farthest thing to the left is our smallest value. So it's a negative infinity. It's a sideways eight. And if you notice here, positive and negative infinity always have a parentheses with them. Okay? So I'm not going to include the lowest point, which is negative infinity. And I'm going to go all the way up to one. And in this case, I am going to include it so it has a bracket. X is greater than negative six. Okay? This means not included, and this means include. So X is greater than negative six. It's going to go to the right of six because it's bigger than negative six, but it's not going to include it. So this time, I have a parentheses here at negative six, and it's going to go this way forever up to positive infinity. So the interval notation, we start at negative six, and we don't include it. So it has a parentheses, comma, the largest value, which is going all the way up to infinity, and you never include infinity, so it'll be a parentheses. You can't include infinity because you can't ever get there to say, this is exactly where I'm stopping because there's always one more. All right. What happens, though, if we have a compound? This is what we call compound inequality. What are we going to do here? Well, we're going to really think about as negative seven is less than X, or if you want to, you could write the X first, and since it's open to the X, I rewrite it open to the X in negative seven, and then I also have X is less than four. It doesn't include either one of them. So down here at negative seven, X is greater than that, so it's going to go this direction, and it's going to be a parentheses because I'm not going to include it, and it's going to go this way forever, and when I have X is less than four, it's going to come here at four, not include it because it's a parentheses, or less than, and it's going to go this way forever. And if you notice, in between here, in between my negative seven and my four, there's both a green and a purple line, so I'm going to shade that in because that's really where my answer is. My answer is really from here to negative seven, and up here to four, and everything in between. Over here, those numbers are less than four, but they're not greater than negative seven. Over here, all these values up here, they're greater than negative seven, but they're not less than four. The only thing that satisfies both of those inequalities is what's in between. So how do you write that interval notation? Well, what's the smallest value that's in black here? It's a negative seven. And do we include it? No. And then we go all the way up to four, that's the biggest value in our interval, and do we include it? Now, the inequality says we don't, so it would be a parentheses. So we need to learn some rules before we start working with inequalities and trying to solve for x. So if I add two to both sides, we're starting with four is less than six. If I add two to both sides, four plus two is six, and six plus two is eight. And six less than eight is still true. If I subtract two, that's going to be four minus two, and we want to know if it's less than six minus two. Well, four minus two is two, and is that less than six minus two, which is four? So two is less than four, which is true. So we want to multiply both sides. So four times two is going to be eight. And we want to know if that's less than six times two, which is 12. And yes, that's one's also true. What if I divide both sides by two? So four divided by two is going to be two. And is that less than six divided by two, which is three? Yes, two is less than three. Let's multiply both sides by negative two. Four times negative two is negative two. And is that less than six times negative two, which is negative 12? Negative two is not less than negative 12, so it's false. What if I divide by negative two? Four divided by a negative. I'm going to write it below here, is a negative two. And is that less than six divided by negative two, which would be less than negative three? No, negative three is farther to the left, so it's smaller. So again, we have a false here. There's a rule that says if I have, multiply or divide by a negative, I have to switch the inequalities to make it true. I have a y, and I want to do the multiplication and addition properties just like I do with equations. So I'm going to subtract three from both sides. And I find that y is less than negative five. Well, if I'm going to put that on a number line, I would do something like this, negative five and zero. Just two reference points, that's good for me. And y is less than negative five, so it's a parentheses opening to the left because I've got to go less than, and it's going to go in this direction. And for an interval notation, it's going to go to negative infinity. And it's going to go all the way up to negative five, but not include either one of them. Now, here's the tricky one. I have to divide by negative one. But I'm dividing by a negative and that was one of those rules where we had to switch the inequality. So negative x divided by negative one is x. And four divided by negative one is negative four, okay? So let's make sure that we did this right. Let's get it graphed and then we'll see. So we say negative four and zero. And x is less than or equal to that, so it's a bracket opening to the left because it's got to be less than. So let's pick a number that's in there. Let's say negative five. So the opposite of negative five has got to be greater than or equal to four. The opposite of negative five is five and that is greater than or equal to four. If I tried something over here like zero, the opposite of zero has got to be greater than or equal to four. Well, the opposite of zero is just zero and that is not greater than or equal to four. So we know we did the right thing here. And the interval notation would be negative infinity, because it went all the way down there to the biggest point of negative four. And it includes negative four because of my inequality set and equal to. But never includes an infinity. Finally, I've got a fraction here. I need to fix it up. I need to multiply everything by three. And when I multiply it by three here, it gives me three times negative one-third is going to be negative one times x. Plus three times five, and that's going to give me plus 15. Greater than and then three times four-thirds, which the threes will cancel each other out and we're just left with four. And if I subtract 14 from both sides, I'm going to have to fix my negative x. It's going to be greater than, still carry my inequality. And I have to switch the inequality because I'm dividing my negative one. So that means that x is less than 11. Finally, we have to look at the kind where we have a compound inequality. Now what you do to the center, you do to either side. So this is just three x in the middle, I'm trying to get to x here. So in order to get to x by itself, I'm going to divide that by three. But that means I have to divide the left-hand side and the right-hand side. That gives me negative four less than or equal to x, which is less than three. And on my number line, negative four is the smallest value and three is the largest value. I can include four and it's going to go this direction. I can't include three, but it's going to go this direction. So you can see again that it's anything between the two values. And it's going to be a bracket negative four to three with a parenthesis. One more. What you do to the middle, you do to the outsides. So I need to subtract three from the middle. So I'm going to subtract three from the left-hand side and subtract three from the right-hand side. 15 minus three is 12, less than negative 2y. Less than 25 minus 3 would be 22. Now I have to divide by negative 2. But since I'm dividing, I've got to change my inequalities. I'm dividing by a negative. Since I'm divided by a negative, I've got to switch those inequalities. And divide by negative 2. And that means that 12 divided by negative 2 is negative 6. And this just gives me y in the middle because the negative 2's cancel. And 22 divided by negative 2 is going to be negative 11. And normally, we write our inequalities that are facing to the right. So I really need to write negative 11, because it's the smaller value, which is less than y, because it points to the negative 11. And then it points to the y and the negative 6. So we have negative 11 here, and it's greater than. So it's going to go this direction. And we have y is less than negative 6. So we have negative 6 down here. And it's less than that, it's going this direction. And we have anything in between. If we want to go back up here, we could look at this and just think about this. You could just disregard this if you wanted to. And you could say y is less than negative 6. So that's what we did in red. And y is greater than negative 11. And that's what we did in green. And then it's everything in between. And that means that our smallest value is negative 11, not included, in parenthesis, comma, negative 6 is the largest value on that interval. And it also has a parenthesis.