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Hello and welcome to another video.

In today's lesson, we'll be doing a second lesson on inequalities and the Cartesian plane.

Now, remember, I'm Mr. Maseko and before you begin this lesson, make sure you have a pen, a pencil, and something to write on.

A ruler will also be really useful in today's lesson.

So pause the video here and go get those things.

Okay, now that you have those things, let's begin today's lesson.

Look at these diagrams and then tell me what's the same, what's different in these representations.

Pause the video here and give this a go.

Okay, now that you've given this a go, let's see what you've come up with.

Well, the first thing, if we look at this, think back to what we did last lesson.

This is an inequality and this diagram is showing this inequality graphically.

So this inequality is being shown on this diagram.

Well, how do we know? Well, we can see we have a line on x equals two and we know that x has to be greater than two.

So if you look, all our x values are bigger than two.

And then this line we haven't seen yet, so what does that mean? Well, that means, if you think back what you did in year seven, that means less than or equal to.

So any time we have a combination of the inequality sign and an equal sign, so that is less than or equal to.

So this inequality says x is greater than two but is less than or equal to seven.

So we look at the line x equals seven.

That vertical line.

If we look here, x is less than or equal to seven.

Now, if you look at this diagram, what do you notice about the line x equals two and x equals seven? Well, you should notice that that line, x equals seven, is solid.

Whereas, the line x equals two is a dashed line.

Now, why do you think that is? Look at the inequality signs we've used.

Why do you think the line x equals two is dashed and the line x equals seven is solid? Exactly, that's because x is either less than or equal to seven.

So x can also be seven.

So we draw the line, our boundary line solid to show that x can also be that value because x is less than or equal to seven.

Now, this representation is something we haven't seen yet but it is also saying the exact same thing as this and we'll explore this more in today's lesson.

Now, if you look at this, these two are showing the same thing but this inequality is not showing that because this inequality is talking about x, whereas these two diagrams are describing boundaries for y.

Let's look at this.

So we've seen this already.

This blue line represents the same thing as this inequality and also this representation.

Now, this representation is an inequality, this is an inequality that is shown on a number line.

So what do you think the shading means? Why is one circle not shaded in and the other circle shaded in? Think back to what the inequality is telling us.

So what is the inequality telling us about x? Right, it's telling us that x is greater than three but it's less than or equal to seven.

So what do you think the shading and the not shading in signifies? Exactly.

It shows that x can also be seven.

Remember, x is less than or equal to seven.

So it's less than or equal to.

So when the circle on the number line representation of the inequality is shaded, that's just showing that x is less than or equal to that value.

So x can also be seven.

But the fact that it's not shaded in, what does that mean? It means that x cannot be three.

So x cannot be three but x can be seven.

And on this number line, on the graphical representation on this slide, that's shown using a dashed line to tell us that x can't be three.

So x can't be on the line x equals three.

But if you look, there's a solid line on x equals seven 'cause that shows us x can be seven.

So remember, x is greater than three but is less than or equal to seven.

So inside that blue region that's shaded in, that's all the values of x we can have.

So these three images are all showing the exact same thing in different ways.

There's the inequality and this is the inequality shown on a number line, and that's the same inequality shown on a coordinate grid.

So let's look at this further.

So pause the video here and see if you can write down the inequality that describes the red region.

And then show if you can show that inequality on a number line.

So pause the video here and give that a go.

Okay, let's see what you've come up with.

Let's look at our boundaries.

Well, we've got the line y equals three and y equals -3.

But what do we have? The line y equals three is dashed and the line y equals -3 is solid.

Our y values are all greater than -3 but what does a solid line signify? Exactly, so our y values have to be greater than or equal to -3 but the line y equals three is dashed, so what does that mean? Y cannot be three, so y has to be less than three.

So y lies within that boundary but it can also be -3 because it's greater than or equal to -3.

So how would we show this inequality on a number line? Well, it's between the boundaries of -3 and 3.

And we're looking at our y value.

Do we have to shade anything in? Yes, we need to shade in the circle underneath -3 because that shows that y can be -3 but it can't be 3.

So if we look at all the integer values of y, what can the integer values of y be? What are integers? Whole numbers.

So what whole numbers can y be? Well, y can be -3, it can be -2, -1, zero, one, two.

Can it be three? No, it can't be three 'cause it has to be less than three.

So these are all the integer values that y can be because it's greater than or equal to -3 but it has to be less than three, so three is not included in the lists.

So at this point, here's an independent task for you to try.

Pause the video here and give this a go.

Okay, now that you've tried this, let's see what you've done.

For that first one, we're going to show those inequalities on a number line.

Well, our boundaries for the first one is from -2 up until five.

And we're describing x.

Let's get our circles.

Well, what's that inequality telling us? X is greater than -2 and less than or equal to five.

So that means the circle under five has to be shaded in.

The second one between one and three, we're describing y.

And y is greater than or equal to one, so one is included and less than or equal to three.

So three is also included.

In that last one between five and eight and we're describing x.

Well, x is less than or equal to eight, so that is shaded in but five is not shaded in 'cause x has to be greater than five.

Now, these inequalities, this would be x being greater than four but less than or equal to 10.

This is y being greater than or equal to -4.

And less than two.

Remember, shaded in means that it can also be equal to that number.

Good.

So for our explore task, here are two inequalities.

Can you list all the coordinates with integer values of x and y that satisfy both those inequalities? Pause the video here and give this a go.

Okay, let's see what you have come up.

What's the first thing we had to do? Well, the first thing that you should have done or looked to do was to draw your boundaries onto your coordinate grid.

And we'll start with the x boundaries.

Well, we know that x is greater than -2.

So if we go to the line x equals -2, so that's the line x equals -2, why did I draw the line dashed? Good, because x has to be greater than -2.

It can't be -2.

And because the line x equals three and we also draw that dashed because x is less than three but it can't be three.

And we do the same thing for y.

We go to the line y equals zero.

Well, this is the line y is equal to zero.

Coincidentally, if you notice, the line y equals zero is the x-axis.

That's the x-axis.

And that is y is equal to zero.

Think about why that is.

And then if you look at the line y is equal to two, we're going to draw that solid as well.

Y equals two.

Why did we draw both those lines solid? Good, because y can be greater than or equal to zero, so it can't be zero and less than or equal to two, so it can also be two.

So we are looking at coordinates in this region so they can be on the line y equals two but can this coordinate be? Well, no, that coordinate can't be because it's on the line x equals three and we know that we're not allowed to be on that line.

So these are all the coordinates that you can have that have integer values of x and y.

So thank you very much for participating in today's lesson.

If you want to share your work, ask your parent or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak.

I'll see you again next time.

Bye for now.