Common Core Algebra I.Unit 2.Lesson 12.Interval Notation.by eMathInstruction

There's going to be many times and we've seen this already with inequalities. There's going to be many times when we're going to want to look at a section of the real number line. We're going to want to be able to summarize a chunk of the real number line, either a horizontal number line, X, or a vertical number line, Y and we're going to want to be able to talk about these chunks of the number line using some kind of notation. So far what we've used is what's known as inequality notation. Sometimes known also a set builder notation. Today, what we're going to look at is something called interval notation. Now, in our interval notation is not per se included in the common core. But then again, neither is any type of really inequality notation. They don't really get into that in the common core standards. But New York State and other states allow use. 

It's kind of cool, and a lot of students prefer it over what we've seen before. So I think you'll like it as well. All right. Let's jump right into it. Exercise one says for each of the following, graph the portion of the number line described by the inequality, then write the equivalent using interval notation. So we've seen an inequality like this. And remember, what this really says is that X is greater than or equal to negative three. And X is less than or equal to 5. So we want all the numbers that fulfill both of those two things. Well, if we start at negative three, and we go all the way to 5, and remember we fill those in with solid dots because that's the deal. When we have the equalities there, then that's the number line. Now let me show you how you do that in interval notation. It's really great. You write down where you start, you put a comma, you write down where you stop, and because the negative three is included, you put a bracket. And because the 5 is included, you also put a bracket. It's pretty much that simple. Where you start the leftmost point, where you stop, the rightmost point, and then brackets to include those endpoints. 

Now let's take a look at letter B in letter B, we've got a scenario where we've almost got the identical situation as before, X is greater than negative 6, while at the same time X is less than four. The only difference here is that the negative 6 is not included, so we've got something like that, and the four is not included. So we've got something like that. I'm going to try something a little bit different here. I'm going to see if maybe I can just use my prefab tool of hat looks really quite nice. I should have been using that all along. Anyway, that's the chunk number line that we now have colored. Now the interval notation works exactly the same. We start at negative 6, whoops. That's not really what I want. Apparently it looks like I start at negative one. I start at negative 6, I end at positive four. But now somehow we can't use the brackets anymore. We have to use some symbol that says don't include the negative 6. Don't include the four. And we do that by using parentheses. 

All right? We're going to summarize this all eventually, but the gist is, write down where you start, write down where you stop, leftmost, rightmost, and then you use brackets, known as a bracket, use a bracket if it's included, and you use a parentheses if it's not included. Or an open circle. That's a lot to take in, but we're going to get a lot more practice in just a moment, okay? And I think what you'll do is you'll like it a lot more than using the less than symbols. I'm going to clear out the text so pause the video if you need to right now. Okay, let's do it. Moving on. Let's take a look at one that's kind of mixed together. That's sort of helpful. This one says that X must be greater than negative four and X must be less than or equal to 8. So we're going to put a circle on the negative four, we're going to fill in a dot on the 8 and then we're going to color in the number line in between the two. Okay? Now watch how consistent the interval notation is. 

Negative four is where we start 8 is where we stop, we're not including the negative four, so we have a parentheses, we are including the 8, so we have a bracket. There it is. So we're just talking about chunks of a number line. Sets of continuous real numbers that go in this case from negative four to positive 8, not including the negative four, but including the 8. Now, let's take a look at D D seems a little bit more tricky, right? It says X is greater than or equal to four. All right, so in other words, we've got the four and it's filled in, but now we're just coloring in, whoops, we're just coloring in all of this. And in fact, we just want an arrow there at the end. Um. So how do we do this? Well, we know that we begin at four. Okay, the question is, where do we end? Well, we actually don't end, but we need a number here. So what we're going to do is we're going to put in our friend infinity. Okay? Is that really is in a certain sense the rightmost number? Notice how I said in a certain sense, though. 

We're going to put the brackets, the force included. But we're going to put a parentheses because we never can reach infinity. All right? Never reach infinity. So since we can't reach infinity, we can't include infinity. All right? That's a little bit more confusing. To make matters worse, let's look at a lesson in situation. In letter E, what we've got is we've got X is less than 5. Okay? So we've got that open circle on the 5. Then, of course, we've got a line now that's going to extend to the left, okay? And then our arrow. That's confusing. A lot of people want to write down 5 comma something. Can't do that. We got to move from left to right. Move from left to right. So technically speaking, the first number that we have in our interval is negative infinity. And our inner vocals all the way up to 5. Okay? Way out here is a negative infinity. Everything's colored from there, all the way up to the 5. I'm not leaving out these chunks I just lifted my pen. Just like positive infinity, negative infinity is never ever included. And in this case, neither is the 5. 

All right? Now, look carefully at F, F is tricky. You know, we see this, and we read it as negative four is less than X, but it may be help more helpful to think about it as X is greater than negative four. So hopefully you recognize that these are the same thing. Okay? So that's not so bad then, right? If we know that, then I can put a circle on the negative four, I can then draw my line with an arrow on it. Like this, that almost looks like an arrow tip. And now we're back to something similar to letter D, right? We start at negative four, we're going to go all the way to infinity, but we're not going to include the negative four, and we never include the infinity. That's essentially it. Okay? So please pause the video now, think hard about what I did and copy down anything you need to. Okay, I'm going to scrub out the text. Let's keep going. So interval notation. 

Let's summarize this. The bracket means that we include the parentheses means that we don't include, and we never, ever include negative infinity or positive infinity. So we always have the brackets around these two guys. All right? So it's pretty easy. And what's great is then you don't have to worry about which way these symbols are going, you know, a lot of students will get confused with stuff like this. This is the most common thing I get. This drives math teachers. Crazy. Right. Because that just literally almost doesn't make sense. That means X is greater than negative three and X is greater than 5. Don't get me wrong. There's plenty of numbers that are greater than negative three and greater than 5. It just happens to be a bunch of numbers greater than 5. Think about it a little bit. But with the interval notation, all you have to know is where you start. Where you stop, and then whether you use those parentheses or brackets.

Anyhow, I'm going to clear all this out, so pause the video if you need to. Okay, let's do some more problems. Letter exercise number two, which of the following represents the equivalent interval two, the following. Pause the video now, take a very short amount of time and answer this question. All right, so hopefully this is a quick and easy multiple choice question. The correct answer is this one, right? The reason why is that the negative 12 is included due to the equal sign, so we include negative 12, but because of just the strictly less than sign here, we don't include. We don't include our friend four. All right? So to include, we use the bracket to not include we use the parentheses. So simple enough. We are going to use interval notation all over the place this year. We'll also use the other notation because depending on what school you go to, perhaps even what state or country you're studying this curriculum in, you may or may not use interval notation. 

It is commonly used in higher level math because it's so easy to write intervals this way. But still, you never know what your teacher wants you to use. The state of New York, at least, will allow you to use either one of them on answering a free response question on a test. Okay, I'm going to clear this out. Let's keep going. The whole focus today is going to be on interval notation, but the beautiful thing is this allows us to review a lot of other things, including solving inequalities. So let's take a look at exercise number three. It says solve the inequality below for all values of X, graph the solution on the number line given and state the solution set using interval notation. All right. So let's solve this inequality. 

Now the first thing I'm actually going to do is just use the properties of real numbers specifically the commutative property to flip flop that subtraction into an addition of negative four X plus 12, right? This is the same as 12 plus negative four X so we use the commutative property to write it this way. Now I only do that because for me then I can use one property of inequality, which says I can subtract anything I want from both sides, no problem there. And I'll get negative four X is greater than negative 12, I'm going to rewrite that right up here. Negative four X is greater than negative 12. And now remember, maybe the most important property of inequalities is that when I divide by a negative, I have to flip flop that sign also remember that a negative divided by a negative is a positive. So here's my final answer. X is less than three. 

Now if I graph that, what do I have? Well, I have a circle on the three, right? I have a line which extends this way with an error. All right. And I write that interval notation. I remember I have to pick the first number that gets colored, and that is negative infinity. Comma, the last number, which is positive three, and neither one of those, neither one of those are included. We don't include the negative infinity because we can't ever get there. We don't include the three, because there is no equality on that inequality. All right. But simple enough, right? Little review of how to use properties of real numbers, commutative property there. Properties of inequality to solve that. Okay, I'm going to clear out the text. We'll go on to the next problem. Here it goes. All right, let's move on. 

Exercise four. Two inequalities have solution sets given in the interval notation below. So one inequality has this solution set. And one inequality has this solution set. Letter a says write an interval that represents all values that are solutions to both inequality. So we're talking about compound inequalities here, right? A compound inequality, putting two inequalities or two intervals together. Draw a number lines to help you think about the solution set. So think about this for a little bit. Let's draw some number lines beautiful there. This will be number one. And I'm going to put zero on here just for a little bit of reference. Negative three might be about here and two might be about here. We know from the interval notation the bracket here that the negative three would be included, right? But the two wouldn't be. So then this is lovely because now I'm working in blue. 

Let me actually tell her it didn't read. That now, along with the closed dot, becomes my solution set. Let's graph the second one. This will help us actually for part B as well. Again, let me put my zero on. Four is somewhere out here. Now in this case, neither the zero, nor the four are included. And again, let me go ahead and put my shading in with red. All right, so now we want and remember how and works. We want all the numbers that are included in both. So it's going to be this overlapping region right here. Now, you might say to yourself, right, you might say, well, okay, that's just zero to two. But you have to be careful. Zero is not included here, and two is not included here. So technically speaking, the overlap is zero comma two, right? Where neither are included, right? The two is not included because it's not included up here. And the zero is not included because it's not included down there. 

So there's our final answer. Now letter B says, write an interval that represents all values that are solutions to either of the inequalities or now remember, or is going to be true as long as at least one of the two things is true at least one. They can both be true. That's okay. So this is kind of cool. Let me do it in a different color. Let me go, that's not what I want. Let me go green. So this is where it starts. And this is where it ends. Because any number that's between these two is either in this, or it's in this one, or it's in both. And it's okay to be in both. There's nothing wrong with it being in both. We talked about that, right? This is what's known as the inclusive or. So both can be true. Now let's think about that for a moment. Going back to blue, we start at negative three, and we go all the way up to four. The negative three is definitely included because it's colored in. But the four is not included because it's not shaded in. 

So there is our compound inequality with ore. Okay, it's kind of cool. And an ore. Very important ideas. All right, I'm going to clear this out, so pause the video now. This is one that's important. You might want to spend a little time thinking about it. Okay, clear it out. All right, exercise number 5. At a hydroelectric plant, pump one is on for all times in the following interval. Pump two is on for all times in that interval. Which the following represents all times when both pumps are on. Pause the video now and think about this problem. See what you can come up with. All right, let's go through it. So we could do this exactly the same way we did the last one. All right? If both pumps are on, both pumps are on. That's really an and scenario, right? We need them to overlap. So how does that look exactly? Well, for pump one, let's start our number line at zero, and let me put 8 right here. Pump number one starts at zero and goes all the way to 8. Goes all the way to 8. But then there's an open circle. 

On the other hand, pump number two, so here's number one. Here's number two. Starts at four, right? That's right here and goes all the way to 18. That's right here. Four gets colored in. 18 does not. All right, so we want to know all the times when both pumps are on. Okay? Well, they're both on it four. That's the first point where they're both on. And then pump one shuts off clearly at 8. So they're both on in this interval. Now, that interval includes the four, right? But it doesn't include the 8. So if we were to write it in interval notation, it would look like that. But in our inequality notation, it would be T is greater than or equal to four while being less than 8. And that is choice one. All right? 

So that's interval notation. It's really just a way of showing what we're talking about. It's not math per se, it's the way that we're writing math. And that's important, because it's a commonly accepted notation, and a lot of you will like it when we get into what are known as functions and their domains and ranges. For now, it's just a way of really summarizing solutions to inequalities and compound inequalities. All right. Pause the video now if you need to, and then I'm going to clear out the screen. Okay, here we go. Let's finish up the lesson. So there are going to be many instances where you want to describe a chunk of the real number line. Oh, I'm talking about all numbers that go from 5 to 18 or from negative two to 7. You're going to want to do that. 

All right, there are many different ways of doing it. But one of the most convenient and easiest ways is interval notation, because interval notation simply needs to know the number you start at. The number you stop at, and whether those numbers are included or not. All right, and that's what we learned today. So I want to thank you for joining me for another common core algebra one lesson by E math instruction. My name is Kurt Weiler, and until next time, keep thinking and keep solving problems.