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Hello everyone, it's Mr. Millar here, Welcome to the third lesson of the week on inequalities.

And in this lesson, we're going to have a look again at inequalities and substitution.

First of all, I hope that you're all doing well.

Let's have a look at the try this task to start.

So, here we've got a number line here with p, q and r.

So you can imagine a number line with numbers here.

For example, you could have one, two, three, so p is just more than three, r is equal to one, and you've got minus one, minus two, minus three.

So, the task for you to do is how many different inequalities statements can you write about p, q and r? So, Zaki has done one example.

He said that p is greater than r because as you can clearly see, p is to the right of r on the number line.

So, how many more inequality statements can you think about with these three unknowns? Have a think, pause the video now.

Okay, so there's lots that you could have thought of.

You could have said that p is greater than q.

Or the other way around.

You could say that q is less than p, and you could also say that r is greater than q and vice versa.

You could also say, you can imagine extending r out.

So you could imagine that 2r looks like this.

So you could even make some inequality statements with 2r.

So still p would be greater than 2r.

So yeah, there's any number that you can make.

But it actually turns out that if we think about negative numbers, which is what we're going to have a think about a little bit more in this lesson, that there are more statements that we can write.

So in the next slide, we're going to have a think about this in a little bit more detail.

Okay, so as I said, using negatives allows us to write more inequalities.

So, the first student is saying, "I can show on the number line, what minus r looks like." So even feel free to pause the video, have a think what r looks like, and have a think about any more inequalities that we can write.

Well, as we said, if r is equal to one, then minus r is going to look like minus one.

It's going to go, r is going to go in the other direction.

And the same thing is true for q.

So, if q is equal to minus 2.

5, then minus q will flip it in the opposite direction.

So we'll go to here, this is what minus q is going to look like, and this first one is minus r.

So if we think about negatives here, there's actually a lot more inequalities that we can come up with.

For example, if we can compare minus q to r well actually minus q is going to be greater than r, because it is further to the right on the number line.

Similarly, we can think about other inequalities.

So minus q is going to be greater than minus r for example.

We could say that minus p, which is going to go all the way down here somewhere minus p will actually be less than q.

So think about negatives in terms of flipping the arrow in the opposite direction, allows us to write a lot more inequalities.

I'm going to be having a look at this in some more detail in the independent task.

Here is the independent task.

Let's have a look at the first question and go through it together.

Given that a is equal to minus three, which of the following inner qualities are true? So if you imagine a number line up here, where here is zero, the first one says a is greater than minus four.

So what we're looking at is minus three greater than minus four, and we can think about minus three and minus four on a number line, so minus three here, minus four.

And I know that minus three is to the right of minus four, so it's greater, so this is true.

And the second one is where we need to use our knowledge from the connect task.

So minus a is going to be minus minus three.

And if we think about that, if we remember the connect task, taking the negative of a negative number is going to make that positive.

So minus a is equivalent to three.

So three is less than minus four.

Well, obviously, three is going to be up here.

We know that three is greater than minus four, so this one here is going to be false.

Okay, so using this concept, can you complete the rest of these tasks? So pause the video for a couple of minutes to see how far you can get.

Okay, great.

So answers are coming up on the next slide.

Okay, so here are all of the answers and why don't we just go through one of these questions together? Let's go through the last one then, minus p and minus q.

Well, minus p is just going to be minus 10 and minus q is going to be minus minus 10, which is equal to 10.

And therefore, we use the inequality sign less than because minus 10 is less than 10.

Okay, let's have a look at the explore task to finish the lesson.

Okay then, so let's have a look at this explore task.

So, if we know that 2a is less than b, are the statements below always, sometimes or never true? Now, to help you out with this, what you need to be doing is thinking about different values of a and b, which satisfy the inequality 2a is less than b.

Now for example, you could say that you could have a is equal to one.

So 2a would be two times by one, which equals two.

And a value of b that would make that work might be b is equal to three.

And you could also have negative numbers as well.

So for example, if you had a is negative two, then 2a would be negative four.

And so, you could have a value of b as negative three, for example.

So anyway, this is the general idea.

You need to find different values of a and b, which satisfy 2a is less than b and then, you need to decide whether the statements below are always, sometimes or never true.

So for example, if you had a is equal to one and b is equal to three, and you put those numbers into the first inequality, does that inequality hold true as well? So pause the video and investigate this task to see if you can decide whether these statements are always, sometimes, or never true.

Okay, so let's go through these then.

And you should have found for the first one, that whatever values you tried, the statement was always true.

So the first one was always true.

And the last one actually, a equals b, that was never true.

So you could never come up with values of a and b that satisfies 2a is less than b such that a was equal to b.

That's just impossible.

But the other two, that ones that both involve negative numbers are actually both sometimes true.

So there are some examples where it is true and some examples where it's not true.

So even feel free to pause the video now, in case you haven't got this, to find an example where the statement is true and an example where it is false.

Okay, so well done if you found it, let's go through it first of all.

So, this first one here, it is actually true if a and b are both positive.

So if we had, as an example, a is equal to one and b is equal to three, then minus 2b would be minus two times by three, which was minus six, and a is equal to one.

So minus six is less than one, that is true, but it is false if actually a and b are both negative.

So, if we had a is equal to minus two, and b is equal to minus three, then minus 2b would be minus two times by minus three, which is six, and a is minus two, so that statement would be false, if a and b are both negative.

And the final one again is sometimes true.

Minus a is greater than b and even feel free if you haven't got this so far to have a think of some values of a and b that are true or false.

Well, well done if you found this out and again, if a and b are both positive, then this statement is false.

So if we had as before, a is equal to one and b is equal to three, then this would be false, because minus a would be minus one, which is not greater than b, which is three.

So that would be false.

But if you had some negatives in here, then you would find that this is true.

So for example, if you had a is equal to negative five, then you could have b is equal to negative four for example, because minus a would be minus minus five, which is equal to five, and b is just minus four.

So yeah, that would definitely work.

Okay, so that is it for this lesson.

A little bit of a tricky explore task, but the whole point is just to get you thinking about your negatives here in the context of inequalities, because in the next lesson we're going to be taking this a step further and thinking about solving some inequalities.

So anyway, hope you've enjoyed today's lesson and I'll see you next time.

Have a wonderful day, bye-bye.