Hello, amazing math people. We're going to look at inequalities in one variable. There are many different types of inequalities. Some of them are less than, less than, or equal to some number, greater than, greater than, or equal to some number, or not equal. When we look at these to graph them, we're going to have less than, greater than, and not equal to on a number line graph. All being open dots or sometimes thought of as hollow dots are less than or equal to or greater than or equal to and just our plain old equal are all going to be solid dots. So if we take just a moment and look at a few examples, X greater than two, we'd have our number line and we'd have an open dot at 2:00 and we would think about shading all of the numbers that are bigger than two or greater than two. So we're going to shade off to the right. Sometimes students think about if it's greater than and the variable comes first. This is an arrow pointing to the right, but it's important that the variable comes first. The next one we want to look at is something like X less than or equal to -1. Once again, we're going to have our number line. We want numbers that are now less than or equal to, so we're going to use a solid dot. Once again, if we think of this as an arrow, the arrow's shading to the Oregon pointing to the left. We can also have the variable occur between 2 numbers, like -2 less than or equal to X less than or equal to 1:00. We'd start on a number line. We'd look at each of the inequalities separately. Less than or equal to solid dot on -2, less than or equal to would be also a solid dot on one. But the X is in between those values, so we would shade in between. Now we actually want to also talk about some notations. We have set notation and we have interval notation. Set notation means that we're going to look at the set which is used with braces of all the X values such that X is greater than two, or the set of all X that X is less than or equal to -1 the set of all X such that -2 less than or equal to X less than or equal to 1. Now that can be kind of cumbersome if we have a lot of notation. Interval notation is a much shorter notation, especially as problems get more complex. Interval notation is what I'd really like us all to be using. And the way interval notation works is if it's an open dot, we're going to use a parenthesis and we start our shading at the number 2 and we keep going until when, well, we always keep going. So we keep going till Infinity. That eight on its side is called Infinity. Here. We would look and we'd say, where does it start being true? Well, it starts being true at negative Infinity for its shading. Where does it stop being true at -1 and is it a closed or open dot on -1 It's a closed or solid dot. So we're going to use a bracket -2 less than or equal to X, less than or equal to one. Once again, we look at it and we say where is it starting to be shaded? And that's at -2 and it was a solid dot. So it's going to be a bracket. Where is it stop being true at 1:00? It's also a solid dot, so we're going to use a bracket. We also need to look at things like the addition principle and the multiplication principle. Now these work just like in a regular equality, with the exception of if we're doing multiplication or division by a negative value. So if I had X + 2 greater than 1 and I wanted to get X by itself, we would say we're going to subtract 2 from one side, and if we subtract 2 from one side, we subtract it from the other. So we'd get X greater than -1 set notation X such that X greater than -1 interval notation. Well, it's going to start being true at -1. It's going to stop being true at Infinity. We always use parentheses for Infinity because we're never equal to Infinity. If you can give me a number and say that's Infinity, I'm going to add 1 to it and have something bigger multiplication. If we had one seventh X less than 3, multiply each side by 7. So we'd have one 7th X * 7 less than 3 * 7, the one 7th and the seven are going to cancel out. So we'd get X less than 21. So here set notation X such that X less than 21 interval notation. It starts being true at negative Infinity, stops being true at 21. It's a parenthesis because it was just less than. Now here's the one that's a little trickier, and it's going to be if we have a negative sign. So let's say -3 X greater than or equal to six. When we multiply or divide by a negative the entire inequality, we have to flip the inequality sign. So once I divide or multiply the entire equation through by negative, I have to flip the inequality so I get X less than or equal to -2. So here set notation X such that X less than or equal to -2. Interval notation that starts being true at negative Infinity stops at -2 with a bracket because it was or equal to. Now let's look at that example one more time, and let's just do it a little bit differently. If I thought about adding 3X to each side, if I had -3 X and I add 3X to each side, I'm going to get zero greater than or equal to 6 + 3 X. Now I have a positive X value, so let's get rid of the six. So we're going to subtract 6 from each side and we're going to get -6 greater than or equal to 3X, divide by 3, so -2 greater than or equal to X. This is the equivalent of saying X is less than or equal to -2, which is indeed what we got up here when we multiplied or divided through by a negative. The last thing on this clip I want to watch show you is I want to talk about domains inside square roots. So if I want the domain of some function and the function is sqrt 4 X -7 inside of a square root has to always be a positive number or zero. So we're going to say four X -, 7 has to be greater than or equal to 0, and we're going to solve for X. So we get X has to be greater than or equal to 7 fourths. Set notation X such that X greater than or equal to 7 fourths interval notation. It starts being true at 7 fourths, so bracket 7 fourths stops being true at Infinity. You've done great. Have a wonderful day.