Lesson video

Hello and welcome to another video.

My name is Mr. Maseko.

In today's lesson, we'll be looking at inequalities and the Cartesian plane.

Remember, before you begin your lesson, make sure you have a pen or pencil and something to write on.

Try this activity and see what you come up with.

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

Well, we are told that the x-ordinate must be greater than two but it must be less than seven.

So what could we have? Well, a number greater than two but less than seven, you could have had yeah, three.

It's greater than two and less than seven.

Well, for the y-ordinate, it must be greater than -3 but less than 3.

So you could have had, yeah, you could have had one.

Greater than -3 but less than 3.

There are many more combinations you could have had.

The numbers that you could have had for your y-ordinate are what? Well, you could have had -2, -1, zero, one, two.

So all of these numbers could have been your y-ordinate.

What about for your x-ordinate? Greater than two but less than seven, so three, four, five, or six.

Now, how many different combinations are there? Can you figure this out? Well, let's explore and see how many different combinations we can make.

This diagram here is a representation of that first problem that we've just gone through.

Now, how do I know this? Remember, last lesson, we talked about horizontal and vertical lines and naming them.

Well, let's look at these horizontal lines and see what we have.

Well, this line, the top horizontal line that we have, on that line, the y-ordinate is always three.

So that line is y is equal to three.

And what about that bottom horizontal line? Well, on that one, the y-ordinate is always -3, so y is always equal to -3.

Now, remember the rule that we had for our y-ordinates.

Well, our y-ordinates had to be greater than -3 but they also have to be what? Less than three.

So any y-ordinate above the line y equals -3 and below the line y equals three.

So the same thing can be done for your x-ordinates.

Now, we look at our horizontal lines 'cause remember, with our horizontal lines, no, this is our vertical lines, because remember, the x-ordinate is always the same on a vertical line.

So our first vertical line is what? That is x is equal to two, 'cause x is always two and that second vertical line, well, that is x equals seven.

What was the rule for our x-ordinates? Well, the x-ordinate always had to be greater than two but it had to be less than seven.

So any x-ordinate after the first line and any x-ordinate before the second line works.

Now, there is another way we can write this instead of drawing it out as a diagram like this.

We can write this as an inequality.

So we're going to look at inequalities.

Well, what are inequalities? And with those signs, you have your greater than and your less than signs.

Well, what rules did we have? Well, we said x had to be greater than two.

So remember, x, the line x equals two, our x-ordinate had to be greater than two.

So our x-ordinate had to be greater than two but they also had to be less, our x-ordinate had to be less than seven.

Now, because those two inequalities describe numbers that are within a boundary, we can put them together, so we say our x-ordinate, well, it's greater than two but it is less then seven.

Our x-ordinate, so we can read it like this from left to right, two is less than our x-ordinate, which is less than seven.

Or we can say seven is greater than our x-ordinate, which is greater than two.

We can do the same thing for our y-ordinates.

Well, what did we say? Our y-ordinate had to be greater than -3 and our y-ordinate also had to be less than three.

So we can combine this the same way.

We can say our y-ordinate, well, that is greater than -3 and it is less than three.

Okay, now that we've written these as inequalities, let's look back at this diagram and see what we are being shown.

Well, what's the diagram showing? Well, we've got two horizontal lines.

We've already given those lines equations already, so this line was y is equal to -3 and y is equal to three and your vertical lines were x equals two and x equals seven.

So what's that diagram showing? Well, we have a set of coordinates, so integer coordinates where the x and y-ordinates are all integers.

So what are integers? Whole numbers.

So those are the integer coordinates and that satisfy, so that lie within those boundaries that we have been given where y is greater than -3 and less than +3 and the x is greater than two and less than seven.

So if you pick any coordinate in that bundle, the y-ordinate and the x-ordinate will satisfy those two inequalities.

So that is what that diagram is showing us and we'll look at this more in the independent task.

So pause the video here and give this task a go.

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

In that first question, we are meant to write that inequality in words.

What are we getting told? We're getting told that x is what? X is greater, x has to be greater than -5, so x is greater than -5.

How do I know? Because if I read it, x, the arrow is pointing towards the five, so x is greater than -5 and x is what? X is less than three.

X is less than three.

So for that, x is greater than -5 and x is less than three.

The second question, well, that says x is greater than -2 but less than three.

And y is greater than -2 but less than two.

And we want to show all of the coordinates that satisfy those inequalities.

Well, if we think back to the picture we saw in the connect task, what did we have? We drew our what? Vertical boundaries.

So when x, so x has to be greater than -2.

So it can't be -2, it has to be greater than -2.

So there's our line x equals -2.

We have to be greater than that.

And there's our line x equals three.

The line x equals three, well, we have to be less than three.

And then for the y, well, y has to be greater than -2.

So there's our line y equals -2.

We have to be greater than -2 and there's our line y equals two and what do we do? We have to be less than two.

So inside that boundary, those are all the coordinates that satisfy both these inequalities.

Now, can we pick a coordinate on any of those lines? Well, no because y can't be two, it has to be less than two, so we can't have a coordinate on the line y equals two.

It's the same thing for all the other lines.

We have to be inside those boundaries.

And so that shaded region is all the coordinates you can have.

So give this explore task a go and see what you come up with.

If you need some help, keep watching the video.

Otherwise, pause the video here in three, two, one.

Okay, if you're still here, I'm assuming you still need some help.

Well, let's think about this together.

What do we have? We're trying to figure out how we can move our boundaries so that we still have 20 coordinates with integer values within the boundaries.

We still need 20 coordinates with integer values.

So what can we do? Well, you can either shift your horizontal boundaries or you can shift your vertical boundaries.

So what if we shift this vertical boundary? So instead of x being less than seven, we say x has to be less than eight.

That's the line x equals eight.

It has to be less than that.

So instead of that, we're going to say x, well, x is less than eight.

What happens to that first boundary? X is less than eight.

What happens to that first boundary? Does it stay the same or change? Pause the video here and give it a go.

Okay, now let's see what you came up with.

Well, if we've moved this boundary to x equals eight, we want to keep 20 integer coordinates.

Well, if we just move this one, then all these coordinates here are now within our boundaries.

So instead of 20, we now have 25.

So what can we do? Well, we could do what? Take this other boundary, other vertical boundary and also shift that one space so that x has to be, so that's the line x equals three and x has to be greater than that.

So we say x has to be greater than three, so now it's only these 20 integer coordinates that satisfy that.

So you could either take both your vertical and shift them one space or both two spaces left or right and you could also do the same thing for your horizontal boundaries.

You could shift them one space down or shift them one space up.

And you would keep 20 integer coordinates within your boundary.

Now, if you want to share your work and show what you discovered and how you shifted your boundaries, ask your parent or carer to share your work on Twitter, tagging @OakNational and #LearnwithOak.

Thank you for taking part in this lesson.

I'll see you again next time.

Bye for now.