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Hello everyone, it's Mr. Millar here.

Welcome to the second lesson on inequalities.

And in this lesson, we're going to be looking at inequalities and substitution.

Okay, so first of all, I hope that you're all doing well.

Let's have a look at the first slide for today, the Try this task.

So here, we've got two students, who are looking at six different statements, and they're trying to work out different values of f.

So the first student says, "My value of f makes exactly two of the statements true." And the second one says, "My value of f makes exactly half of the statements true," i.

e, three out of the six.

So what I want you to think about is, what numbers, what values of f, could they be thinking of? So try different values of f to see how many of these six statements are true.

Pause the video now to have a go at this task.

Okay, so let's have a go at trying out some different values of f and see where it gets us.

So I'm going to start off with f is equal to eight, or you could have tried something different, but here's where I'm going to start.

So the first one, f is less than or equal to six.

Well, this is obviously not true because eight is bigger than six, it's not less than or equal to it.

The next one, six is less than or equal to f, well, this one is actually true because eight is bigger than six.

But the next one is false, the next one is obviously false, and so is the next one.

But the final one, f is greater than six, well, yes, eight is bigger than six, so this one is true.

So how many of these statements are true? Two of them, so actually this first student could have been thinking of the number f is equal to eight because it does make exactly two of these statements true.

So, let's have a think about a different value of f and this time, how about f is equal to six? Well, let's just go through these very quickly.

So the first one, f is less than or equal to six, well, yes, that is going to be true.

And the second one is also going to be true, but the third one is not true because f has to be less than six, f can't be equal to six.

The third one is obviously true, f equals to six.

The next one, not true.

And the final one, f is greater than six, well, f can't be equal to six here, so this final one is false.

So how many of these statements are true? Well, one, two, three, so actually I'm going to to write this out here because three of these statements are true, so I need to put f equals six as one of the statements that the second student could be thinking of, one of the values of f that they could be thinking of.

So actually it turns out that the only two values that work for the second student are f is equal to six or f is equal to minus six, but the first students would be, you could be thinking of any value of f as long as it's not six or minus six.

Anyway, let's have a look at the next task, the Connect task.

Okay, so given that x equals three and y equals two, which of the following inequalities are true.

So three different inequalities to have a go at.

Let's have a look at the first one, and then you can try the next two by yourself.

So x plus y is less than six.

Well, what I'm going to do here is I'm going to substitute in my values of x and y into this inequality to see if it's true or false, so let's have a go.

I'm going to say that three plus two is less than six, and three plus two is five, so five is less than six, and yes, this is true because five is less than six.

You can feel free to pause the video now to have a go at the next one.

Okay, let's go through it then.

So 3x minus y is greater than or equal to seven.

Well, first of all, what does 3x mean? You should know this, but 3x means three times by x, so three times by three minus two, greater than or equal to seven, three times by three is nine minus two is seven.

So we have seven is greater than or equal to seven, and that of course is also true.

And the third one, well, I've got one minus three greater than minus two, just substituting in plus one.

One minus three is minus two, and minus two plus one is minus one, so we have minus two is greater than minus one, and this statement is false.

And we saw this in the last lesson, if we have a number line where we have minus one and minus two, well, because minus one is to the right on the number line as minus two, it is bigger.

So this is the Connect task, hope you find this nice and straightforward to understand.

Let's have a look at the Independent task next.

Okay, so here is the Independent task.

The first question asks you to work out which of the following inequalities are true, so similar to what we did for the Connect task.

The second one asks you to find three different pairs of values of f and g, which make the inequality true.

And the third one asks you to fill in the gaps with a less than, greater than, or equals sign.

So have a go at these questions, pause the video now.

It shouldn't take you any more than six or seven minutes to have a go at as many of these questions as you can.

Okay, so going to the answer slide now, make sure to pause the video in case you haven't completed this.

Here are the answers.

Okay, so here are the answers, and for the first question we got false, true, true.

The second question, you can have any values of f and g, as long as when you add them up together, they are less than or equal to three.

So we can have, we could have f equals two and g equals one.

We could also have negatives or fractions, that is fine.

And the final one, well, we've got less than, less than, and equals.

Let's just go through the third one because it's a little bit tricky where we've got minus 3q, which is minus three times by minus 10.

And that, as you remember, a negative times by a negative actually gives me a positive, so that will be 30, and 3p is three times 10, which is equal to 30.

Okay, that is the Independent task.

Pause the video, if you need some more time to mark your work.

So let's have a look at the final slide of today, the Explore task, and here we've got three different inequalities, A, B and C.

And what you need to do is you need to find pairs of values for m and n so that, in the first example, A, B, and C are all true.

So what I want you to do is I want you to try different values of m and n so that it makes A, B and C all true.

For example, you could try n is equal to five and m is equal to six, and then see if that works for the first example.

You would have n plus three, which would give you eight, and m plus three, which would give you nine, and well, this actually wouldn't work, because eight is not bigger than nine.

So this wouldn't work, and you would need to find a different example.

So, I hope this makes sense, pause the video now to try out different values of n and m that make A, B, and C all true, and the other statements there as well.

Okay, so if you have managed to find different values that work, well done, if you haven't, pause the video and keep going, because it's important that you have a go at this.

But I'm going to go ahead and show you some values that work.

So for the first one, A, B and C are all true.

Well, this works for a lot of different values.

You could try n is equal to five and m is equal to four, and let's just run through this quickly.

This works because for the first one, you would have eight is greater than seven, that works.

And then B, you'd have two times by n, so 10, is greater than eight, which also works.

The final one, n squared 25, greater than m squared 16, so that works as well, so if you have n equals five and m equals four, that will work.

And actually this will work for whatever you chose, as long as n is bigger than m, and they're both positive numbers.

Okay, how about the next one? And while I'm about to note my work in here, I'll give you a clue, in case you haven't worked it out, and the clue is, have a think about negative numbers here.

Okay, so if you have worked it out now, then really well done.

I'll now show you what one possible answer could be.

So if you had an n is equal to minus two and m is equal to minus three, then for the first one, you would have one is bigger than zero.

The next one you'd have minus four is bigger than minus six, also works, but for the third one, you would, it actually wouldn't be true because n squared would be minus two times by minus two, which is four, because a negative times by negative gives me a positive, and then similarly m squared minus three times by minus three would give me nine.

So four is greater than nine, no, this is not true.

Therefore, n is equal to minus two, and m is equal to minus three does work.

And again, this will work for any values of n and m, as long as they are both negative, and n is bigger than m.

So while I'm about my working, you can feel free to pause the video and have a think and see if you can find an example that works for the third one as well.

Okay, and I'll give you the answer in case you haven't found it yet.

And what you need to do is you need to make sure that n is negative.

So for example, minus six, and then m can be anything as long as it's bigger than n, so, for example, it could be equal to four.

It could be anything as long as it's bigger than n, but not bigger than six, because then it wouldn't work.

And let's just show you this quickly, so the first one you would have minus three greater than seven, which doesn't work.

And then you'd have minus 12 greater than eight, which doesn't work.

And in the final one, you'd have 36 greater than 16, so that does work.

Okay, so hope that you found this lesson interesting.

That's all for today.

Next time we're going to have a look at a bit more substitution, so thanks very much for watching and have a lovely rest of your day.

Thanks very much, bye, bye.