Video: Inequalities on a Number Line

In this video, we will learn how to find and deduce the one-variable linear inequality from a specific given interval represented on the number line.

18:00

Video Transcript

In today’s lesson, what we’re gonna be looking at is inequalities on a number line.

And an inequality is something that is not equal to something else. So, we deal with them often in the same way that we deal with an equation. It’s just that we use some different notation. We can see some of that here because, as you can see, there’s quite a lot of strange notation on this page. So I think the first thing we need to do is work out what some of these symbols mean.

So first of all, what we have is our inequality notation. So, we’re gonna start with 𝑥 is greater than 𝑦. And we know that this is 𝑥 is greater than 𝑦 because we got the wide end of our inequality sign by the 𝑥 and the pointy end by the 𝑦. And then, we have 𝑥 is less than 𝑦, and we can see that because we got the inequality sign which is switched around. So, we’ve got the pointy end towards the 𝑥 and the wide end towards the 𝑦. And then, we have 𝑥 is greater than or equal to 𝑦. And we can see here we’ve got the inequality sign which is the same; however, we have this line underneath. And this line is what tells us that it’s all equal to. So, as well as 𝑥 being greater than 𝑦, it could be equal to 𝑦 as well. And then, similarly, we have 𝑥 is less than or equal to 𝑦.

Okay, great. So, this is our inequality notation. So, let’s have a look in practice what they mean, and, in particular, what the difference between less than or greater than or greater than or equal to or less than or equal to is.

Well, let’s have a look at 𝑥 is greater than three and 𝑥 is greater than or equal to three. Now, if we have 𝑥 is greater than three, this means that 𝑥 can take any value greater than three whatsoever. So, it means it could be anything that’s even a slight decimal or fraction above three. However, often in questions where we’re looking at inequalities, it asks us to deal in integer values. So, if we were told to find all the integer values that 𝑥 could represent, then 𝑥 could represent anything from four, five, six, seven onwards because it’s anything greater than three, but not including three.

Now, if we move over to 𝑥 is greater than or equal to three. Once again, it means that 𝑥 could be greater than three. So, it can be any value, any decimal or fractional value, greater than three. But it could also be three itself. But once again, if we were dealing with a question that was asking for all the integer values of 𝑥. Then this time we could have 𝑥 being equal to three, four, five, six, et cetera, with the difference being that 𝑥 this time could represent three itself. Okay, great. So, that’s our inequality notation. But there’s another part of notation that we need to look at as well. And that’s our number-line notation.

So, first of all, what we can see is an open circle with a line pointing to the right, and what this means is greater than. And that’s because if we move right or up a number line, then we get into greater values. And then, what we have is a circle, an open circle here, with the line to the left or an arrow to the left, and what this means is less than. And that’s because if you have a number line and we’re going left or down a number line, then it means the values are gonna be less.

Then, we’ve got a colored-in circle and a line going to the right, and this is an arrow going to the right. What this means is greater than or equal to. And what makes it or equal to is the fact that our circle is colored in. And then, we have a colored-in circle and an arrow going to the left. And what this means is less than or equal to, following the same reason as the one before.

So, great, now that we’ve got the notation and we’ve seen what it means, what we can do is move on to some examples to see how we could use it in practice.

Which part of the shown line represents the set of solutions to the inequality 𝑥 is less than 27?

So, as in the question, we can see that we’ve got 𝑥 and then we’ve got an inequality sign. Then, we’ve got 27. And that inequality sign has the pointy end towards the 𝑥 and the wide end towards the 27. So, what this means is less than. So, we could say that 𝑥 is less than 27. So, therefore, what we’re looking for is the region to the left of 27. Because if we’ve got a number line, we know that anything to the left is down the number line. So, it’s gonna be less than the value that we’ve got.

So, therefore, we can see that our possible values of 𝑥 are gonna be represented by A. So, therefore, we can say the part of the shown line which represents the set of solutions to the inequality 𝑥 is less than 27 is part A. Because, in fact, part B would represent 𝑥 is greater than 27. And that’s because it’s to the right of the 27. So, therefore, it’s all the values that are more than or greater than 27. Okay, great. So, our first example allowed us to interpret some inequalities. Now, what we’re gonna look at is a second example, where we’re actually going to start to use some of our number-line notation.

Which inequality is represented on this number line?

So, first of all, what we can see on our number line is we’ve got an arrow pointing to right. So, this means that we’re gonna be involving greater than. That’s because anything to the right of a value on a number line is greater than that value. Then, next, what we do is we remind ourselves of our number-line notation. And with our number-line notation, if we have an open circle and a line to the right, it’s greater than. But if we have a closed or colored-in circle and a line to the right or an arrow to the right, then it is greater than or equal to. And we can see in our diagram that it is colored in. So, therefore, it’s gonna be greater than or equal to.

So, what this means is, if we’ve got the colored-in dot, like we do in ours, then it’s gonna be greater than or equal to. So, it means it can take the value, so in this case, the value of the negative one cause if it was an open circle, it wouldn’t take this value as well. So, therefore, in word form, we can say that 𝑥 is greater than or equal to negative one.

However, we want to represent it using inequality notation. So, how would we do this? So, using our inequality notation, we have 𝑥 is greater than or equal to negative one. And we know it’s greater than because we’ve got the open side by the 𝑥 and the pointy side by the negative one. So, it’s 𝑥 is greater than negative one. And we know that it’s or equal to because this line underneath our inequality sign. So, we’ve got 𝑥 is greater than or equal to negative one.

Okay, fab. So, we’ve now looked at inequality notation, number-line notation, and how we use them in practice. So, now, what we’re gonna do is have a look at a little bit more notation which brings together our inequality notation and our number-line notation, but in a slightly different way.

So, what we’re gonna have a look at here, first of all, is the double-sided inequality. So, as you can see, we’ve got two inequality signs. And we’ve got 𝑥 in the middle. And what this tells us is that 𝑥 is greater than 𝑦, but less than 𝑧. So, what this tells us is that 𝑥 is between 𝑦 and 𝑧, but it doesn’t include 𝑦 or 𝑧. And, in fact, what our double-sided inequality is, it’s just a combination of two simple inequalities. It’s the combination of 𝑥 is greater than 𝑦 and 𝑥 is less than 𝑧.

So, we’ve got two conditions here that we’ve represented in our one inequality. And if we were to represent this on a number line, the notation we would use for our number-line notation would be an open circle at each end of our line, so an open circle on 𝑦 and an open circle on 𝑧. And then, we would have a line between them that would join them. And what this would tell us is that our 𝑥 could take any value between 𝑦 and 𝑧, but not including 𝑦 and 𝑧, because we haven’t colored in the circles.

And as with the inequality notation that we’ve used previously, we could also include the inequality notation which could also mean or equal to. So, what this would mean would be 𝑥 is greater than or equal to 𝑦, but less than or equal to 𝑧. And once again, it’s just a combination of two more simple inequalities. So, for example, 𝑥 is greater than or equal to 𝑦 and 𝑥 is less than or equal to 𝑧. And once again, if we want to also demonstrate our number-line notation, what we’d have this time would be a colored-in dot above our 𝑦, and then a line, and then a colored-in dot or circle above our 𝑧. And these are joined together by the line. And we can see here that this would be all the values that 𝑥 could take, and they’d be all values between 𝑦 and 𝑧, but also including 𝑦 and 𝑧. And that’s because the circles are colored in.

Okay, great. So, now, we’ve seen this other notation. Let’s move on to an example where we can also see this used.

Which of the following inequalities have been represented on the number line? We’ve got (A) 𝑥 is less than or equal to negative one or 𝑥 is greater than two. (B) 𝑥 is less than negative one or 𝑥 is greater than two. (C) 𝑥 is less than or equal to negative one and 𝑥 is greater than two. Or (D) 𝑥 is greater than or equal to negative one but less than two. Or (E) 𝑥 is greater than negative one but less than two.

So, first of all, what we can see is that our inequality on a number line has two regions. So, we have the region on the left and the region on the right. So, first of all, we’re gonna have a look at the region on the left. Well, the region on the left, we can see that we’ve got an arrow pointing to the left. And what this means is less that because it means anything less than a value cause it’s moving down the number line. And then, we can quickly remind ourselves of the number-line notation because we’ve got a colored-in dot. So, what this means is it’s going to be greater or less than or equal to because an open dot would mean just greater or less than.

So, therefore, we can say that the left-hand part of our inequality on our number line is represented by 𝑥 is less than or equal to negative one. That’s cause our colored-in circle or dot is on negative one, and then we’ve got an arrow to the left. So, then, if we take a look at the right-hand side, then what we’ve got is an open circle. So, this means that it’s gonna be greater or less than, that it’s not gonna be or equal to. And then, we’ve got an arrow to the right, which means greater than. So, therefore, the right-hand side of our inequality on the number line is represented by 𝑥 is greater than two.

Okay, so great. So now, let’s give our final answer, where we can say that 𝑥 is less than or equal to negative one or 𝑥 is greater than two. And it’s this word “or” which is key because it’s telling us that 𝑥 can be less than or equal to negative one or 𝑥 can be greater than two. It cannot be and because you cannot have a value that is less than or equal to negative one and greater than two. That’s why answer (C) would be ruled out. Because here you can see clearly that the word “and” is used; however, what we want is the word “or” to be used. So, therefore, the correct answer would be answer (A), as this is the inequality that’s been represented on our number line.

Okay, great. So, now, let’s have a quick look at the other answers that we’ve been given and why they may be misconceptions and why they’re incorrect. Well, if we take a look at answer (B), the reason answer (B) is incorrect — and this is a common mistake — is that if we look at the first inequality notation, we haven’t got a line underneath our inequality. Which means, therefore, that it’ll just be 𝑥 is less than negative one. Well, if it was 𝑥 is less than negative one, we’d have an open circle on the negative one. But, in fact, we’ve got a colored-in or closed circle there. So, therefore, it’s less than or equal to.

And if we look at (D) and (E), these both are incorrect because (D) and (E) are both — we’ve got here double inequalities. And what these would mean is we’d actually have a region between two values. And the way we’d represent these on a number line is by using two other bits of notation. So, for instance, if we had (D), this would be a colored-in dot above our negative one or on our negative one. And then, we’d have an open circle or dot on our two. And then, we’d have a line joining between them. And what this would mean is that 𝑥 could take any value that was between negative one and two, but also including negative one, but not including two. And for (E), we’d just have an open circle on negative one and an open circle on two and then a line joining them.

Okay, great. So, we answered the question here that has shown where we’ve got two different conditions that need to be met. And also, again, we’ve looked at a bit of double inequalities. But now, we’re gonna have a look at a question that definitely focuses on that double inequality.

Which of the following diagrams represents the inequality 𝑥 is greater than or equal to negative two but less than one?

So, if we take a look at the inequality we have in this question, we can see that it’s a double inequality because we’ve got two inequality signs. And that is, it is an amalgamation of the two inequalities 𝑥 is greater than or equal to negative two and 𝑥 is less than one. And what this is gonna show is a region between two values. So, if we’re gonna want to represent this on a number line, what we need to do is remind ourselves of some of the notation we use.

So, if we take a look at the first inequality sign, we’ve got 𝑥 is greater than or equal to negative two. So, therefore, because it’s greater than or equal to, we’re gonna have a colored-in circle or dot in the line to the right because the colored-in circle or dot means or equal to rather than just that value. And then, the line to the right is gonna be greater than. Then, for the second part of our inequality, we’re gonna have an open circle. And that’s because it’s just 𝑥 is less than one. It’s not 𝑥 is less than or equal to one. And then we’re gonna have a line going off to the left because it’s less than. As we’d already said, this is a double inequality. So, it represent a region between two values. So, therefore, we’re gonna join the lines between our two dots or circles.

So, now, we’ve got something that looks like what we want. Well, we’re gonna represent our inequality on a number line. And we can also see that our colored circle is gonna be on the negative two. And the open circle is gonna be on the one. So, if we take a look at our possible answers, we can see that this is the same as answer (A). Because in answer (A), we have a colored-in circle on the negative two and an open circle on the one. So, what this shows is that 𝑥 can be any value between negative two and one which includes negative two, but doesn’t include one.

So, therefore, we can say that the diagram which represents the inequality 𝑥 is greater than or equal to negative two but less than one is diagram (A). And the other ones are incorrect for various reasons involving the number-line notation. So, (B) has the incorrect dot colored in and the incorrect dot open. In (C), they’re both colored in, and in (D), they’re both open, which will be all incorrect.

Okay, great. So, we’ve now looked at a problem that has a double inequality. Now, we’re gonna move on to our final example. And this is one that’s using a bit of problem solving.

The situation where 𝑥 has to be at least two away from one can be expressed by the compound inequality 𝑥 minus one is less than negative two or 𝑥 minus one is greater than two. Which of the following number lines represents this inequality?

Well, the first thing we need to look at with this question is this wording here, compound inequality. Well, what is a compound inequality? Well, it’s an inequality that has two or more inequalities joined together with either “and” or “or,” like we have here in this question. So, in this problem, our compound inequality has two parts. So, we deal with the left-hand part first, which is 𝑥 minus one is less than negative two.

Well, what we want to do is we want to solve this in the same way that we’d solve an equation. So, we’re gonna add one to each side of our inequality. So, this is gonna give us 𝑥 is less than negative one. Okay, so, that’s the first part of our compound inequality solved. So, we know what our 𝑥-value or our 𝑥-region is gonna be first, and that is 𝑥 less than negative one. So, for the second part of our compound inequality, we have 𝑥 minus one is greater than two.

So, again, once more, what we do is we add one to each side of the inequality to solve. So, we get 𝑥 is greater than three. So, if we put that together, what we get is that 𝑥 is less than negative one or 𝑥 is greater than three. It’s not a double inequality because we’re not looking for a region between two values. So, we’ve actually got two separate parts to our compound inequality. And if we think about our number-line inequality, we’re gonna have two open circles and one with an arrow to the left and one with an arrow to the right. And this is because we haven’t got or equal to.

So, therefore, if we look at our possible answers, we can see that answer (D) is gonna be the correct one. Because we’ve got an open circle in the line to the left at negative one. And we’ve got an open circle on the line to the right at three. And we rule out answer (A) because it’s got two closed circles and answers (B) and (C) because they represent regions between two values.

Okay, great. So, we’ve answered our final problem. Let’s have a look at a quick roundup of what we’ve done in this lesson. So, the key points we have are notation. So that’s what our inequality notation. If it’s a wide end pointing towards it, then it’s greater than. If it’s a pointy end, then it’s less than. And if we have a line underneath, that means it can be or equal to, so it can include that value as well. With our number-line notation, if it’s a colored-in dot, this means or equal to as well as greater or less than.

And, finally, if we have double inequalities, these show a region between two different values. And, finally, we can say that a compound inequality is an inequality with two or more inequalities with an “and” or “or” between them.

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