Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Inequalities and one variable. Inequalities are things that have solutions that are a whole range or grouping of numbers. So if we talk about less than, we want all the numbers to the left of something, or if we have greater than, we want all numbers to the right of something. Sometimes I think about this arrow pointing to the left and the arrow pointing to the right. Less than or equal to just means we're going to include the point, whereas less than doesn't include the point. Greater than or equal to once again, includes the point. Not equal to is a symbol with an equal with a slash through it, open versus closed. If we're less than, greater than, not equal to, we're open, and these are represented with hollow dots on a number line. If we're less than or equal to, greater than, or equal to, or equal to, these are represented with solid or closed dots. So these are solid or closed and these were open. So if we wanted X greater than two, we're going to put 0 somewhere on our number line. So we have our reference, we're going to go over to two and because it's greater than we're going to have an open dot and we're going to shade it off to the right. Now there are two different ways to write this final answer. It's called 1, is called set notation, and set notation says it's the set of all XS such that the condition is true X is greater than two. In this case, that's an acceptable answer. One that's more often used is what's called interval notation. An interval notation uses parentheses for anything that's open and brackets for anything that's closed. So parentheses, and we could use one or the other bracket. We can use one or the other in the group of open. We also include Infinity for parentheses. So in interval notation we started being true at the number two. It wasn't equal, so it was open. So we're going to use a parenthesis, and if we read from left to right, it stops being true. Well, it never does. So we're going to use the Infinity symbol. And because we can never reach Infinity, we use a parenthesis and let you try the next one. This next, 1X greater or X less than or equal to -1. Once again, give yourself A0 reference. This time it's going to be solid and we're going to shade it off to the left. So in set notation, it's the set of all X such that whatever the condition is, is true. So all the X's that were less than or equal to -1. The solution type of notation that we're going to use more often is called interval. We read it from left to right. It starts being true at negative Infinity. It stops being true at -1 and it's going to be a bracket because it included that point. When we look at some more, I'm going to have you try that one. This next one is going to have. Let's actually change this problem. Let's change it to -2 less than or equal to X less than one. If we have -2 less than or equal to X less than one, we're going to give ourselves our zero reference. Again -2's over here, 1's here -2 would have a solid dot, and one would have an open dot, and we want in between. So our set notation would be X such that -2 less than or equal to X, less than one. Our interval notation, reading left to right, it starts being true at -2 and it's included. So A bracket, it stops being true at 1, so 1, and then a parenthesis because it did not include that point. If we look at #7, we're going to solve this. Well, our goal is to get X by itself. So we're going to start by subtracting 2 from each side so we get X greater than -1 and then this turns into what we did a moment ago. If I have zero, I have -1 we're going to put a open dot because it was just greater than. We're going to shade it off to the right because that arrow points off to the right. So set notation X such that X is greater than -1 more useful notation or easier is interval -1 out to Infinity. This next one, we're going to multiply each side by 7, so X less than 21. So when we put, I'm going to actually just go with my reference of 21 instead of putting a 00 would be back here somewhere. So we have X such that X is less than 21 and negative Infinity to 21 parentheses. Now when we multiply or divide by a negative, we have to flip the side. So to get the X by itself, we're going to divide by a -3. When we multiply or divide an entire inequality by a negative, I guess that's an important distinction, the entire inequality. So I'm going to divide this side by -3. I'm going to divide this side by -3. So I have to flip the sign so X is less than or equal to -2. So on. Let's do our graph first. Here's my zero. Here's my -2 solid dot because it was or equal to shaded off to the left because of the arrow. So set notation X such that X less than or equal to -2 interval notation starts at negative Infinity, stops at -2. We use a bracket because it included it. You go ahead and do #10 remember, we're going to get rid of the -3 first, and then we're going to divide by -2 domain problems. When it's a square root, the inside has to be positive to solve over the real numbers, positive or zero. So whenever we find a domain problem using a square root over the real numbers, we're going to take the inside and we're going to say it's greater than or equal to 0, and then we're going to solve. So we're going to add 70 each side. We're going to divide by 4, so as a picture, it starts being true at 7 fourths. It includes the point, so it's a solid dot. We're going to shade it off to the right. Set notation X such that X is greater than or equal to 7 fourths. Interval notation bracket at 7 fourths because it was a solid dot. Infinity parenthesis. You do #12 Thank you and have a wonderful day.