Hey, everyone. We already know how to express sets in set notation, so something that looks like this, but there's actually a much more compact way to represent solution sets in what's called interval notation. Now you might be thinking, why do we need a new way to write sets if we already have one that works just fine? And not only is it going to be quicker and easier, but you're actually going to be asked explicitly to express your solutions to inequalities in interval notation. So, I'm going to show you how to take this set and turn it into something much more compact using parentheses and square brackets based on whether you have a less than sign or a less than or equal to sign. So let's go ahead and take a look here. There are different types of intervals you can have based on whether you're dealing with a less than or less than or equal to sign. So let's first look at our closed interval. So when I have a set like x≤0≤x≤5, because I have these less than or equal to inequality symbols, I am going to write my endpoints in between square brackets. Now you're also going to be asked to graph these intervals, and what that really means is just write your interval on a number line. So whenever I have square brackets on my interval, I'm going to express on my number points with closed circles. So I'm still going to label my endpoints here, and then I'm going to draw a line connecting my endpoints to show that my set or my interval includes everything from 0 all the way to 5, including 0 and 5. So whenever we're dealing with a less than or equal to sign, a square bracket, and a closed circle, that tells me that I'm dealing with a set with endpoints included because it's not just less than 5, it's less than or equal to 5. So let's take a look at our open interval here. So an open interval happens whenever I have something like x<0<x<5. Now you'll notice that these are not less than or equal to signs. They are just less than signs. And when I have that, I'm just going to take my endpoints and stick them in between parentheses. So when I have an open interval, those less than signs become parentheses in interval notation. Now whenever I'm asked to graph this, I'm going to take my parentheses and make them into open circles, still have my endpoints, and still connect my endpoints. But the open circles say to me that my endpoints are not included. They are excluded from my set. So anything I can have any number here from 0.0001 all the way up to 4.999. But I can't have 0 or 5 because they are not a part of my set. So that's an open interval. We can also have a combination of closed and open intervals. So let's take a look at that here. Here we have x≤0≤x<5. So you'll notice that the first symbol I have is that less than or equal to sign, which tells me that I need to enclose that first endpoint in a square bracket because it is included in my set. Now for my second inequality symbol, I just have a less than sign. So that tells me that my second endpoint is just going to be enclosed in parentheses. So that's my interval in interval notation when it is half-open half-closed. Let's go ahead and graph that as well. So whenever I have that square bracket and my endpoint is included, I want to make sure to have a closed circle there and still label my endpoint. And then with a parenthesis, I know that it's going to be an open circle. And then I need to fill in because it's all the way from 0, including 0, up to 5, but not including it. So that's how you express a half-closed, half-open circle. Let's take a look at one last example here. So here my set is x≥3. Now just looking at that, having that greater than or equal to sign tells me that I'm going to be dealing with a square bracket somewhere. But let's actually graph this first and see what's happening. So when I have a greater than or equal to, I know I'm going to be dealing with a closed circle. So I'm going to go ahead and plot my 3 with a closed circle. Now this is going to be fine all the way up until forever, up until infinity. So whenever we don't have an explicit endpoint, so here this just says x can be anything greater than or equal to 3 with no limit, I'm actually going to use infinity symbols to express this. So either a positive infinity or a negative infinity based on what direction it's going in. And whenever we have infinity in our intervals, we're going to treat this as an open bound because you can't get up to infinity. It goes on forever. Right? So it's not a hard endpoint. So to write this in interval notation, my 3 gets enclosed in a square bracket because it is included in my set but then I have infinity which just gets a parenthesis because that is not a hard endpoint. It's not less than or equal to infinity because you can't be equal to infinity. Right? So that's all there is to express sets in interval notation. Let's keep going.
Table of contents
- 0. Fundamental Concepts of Algebra3h 29m
- 1. Equations and Inequalities3h 27m
- 2. Graphs1h 43m
- 3. Functions & Graphs2h 17m
- 4. Polynomial Functions1h 54m
- 5. Rational Functions1h 23m
- 6. Exponential and Logarithmic Functions2h 28m
- 7. Measuring Angles39m
- 8. Trigonometric Functions on Right Triangles2h 5m
- 9. Unit Circle1h 19m
- 10. Graphing Trigonometric Functions1h 19m
- 11. Inverse Trigonometric Functions and Basic Trig Equations1h 41m
- 12. Trigonometric Identities 2h 34m
- 13. Non-Right Triangles1h 38m
- 14. Vectors2h 25m
- 15. Polar Equations2h 5m
- 16. Parametric Equations1h 6m
- 17. Graphing Complex Numbers1h 7m
- 18. Systems of Equations and Matrices1h 6m
- 19. Conic Sections2h 36m
- 20. Sequences, Series & Induction1h 15m
- 21. Combinatorics and Probability1h 45m
- 22. Limits & Continuity1h 49m
- 23. Intro to Derivatives & Area Under the Curve2h 9m
1. Equations and Inequalities
Linear Inequalities
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