Lesson video

Hello everyone, it's Mr. Millar here.

Welcome to the seventh lesson on inequalities.

And today we're going to be talking about manipulating inequalities.

Okay.

So first of all, I hope that you're all doing well and well done for continuing with this topic.

Only two lessons left to go on inequalities and for the try this task, we're going to be having a look at a bar model here and trying to form some different inequalities.

So you can see the bar model on the left, and the different colour bars, each correspond to a letter or a number.

And what I want you to think about is what different types of inequalities you can form.

So for example, I can see that the purple dotted bar, which is represented by a, is bigger than the light blue bar, which is represented by b, therefore a is greater than b.

See if you can find a few more different inequalities and just spend one or two minutes, having a think about what you can find.

So pause the video now and try to write down as many inequalities as you can for me.

Okay, great.

So, I hope you found lots of inequalities to write about.

So you could notice for example, that if you take c and seven, so c plus seven.

So what I'm circling now that is c plus seven, that is bigger than the light blue bar, which is b, it's got a bigger length than that.

So that would be one example.

You could also notice that, b has a greater length than seven.

You could notice that, for example, two lots of a, so two purples has a bigger length than 2b, loads and loads of different possibilities.

And what we're going to have a look at throughout this lesson, and next lesson as well, is that if, for example, I know that a is greater than b, then if I do the same things at both sides, so if I multiply, both a and b by two, so times by two on both sides, then I get at 2a, is greater than 2b.

And I know that's going to be true, because just as if I'm balancing an equation and I do the same thing to both sides, I can do the same thing to an inequality, it will still be true if I do the same thing, that's the both sides.

So let's have a look at this a little bit further in the connect task.

Okay.

So here is the Connect task, given that p is greater than q plus two, are the following inequalities sometimes, always or never true? So yeah, I'll give you a few seconds for you to have a read of these inequalities, and have a think about what you would say here.

Okay.

So before I give the answers, I can, first of all, write these down as bar models to show this, a little bit more easily.

So first of all, here is p, and I can see that it has a greater length than q plus two that is expressing the given that inequality as a bar model.

Now, if I've got at p plus five, then I'm going to have to add a five to the p and if I've got a q plus seven, well, that is the same thing as q plus two plus five.

So really what I'm doing is I'm adding five to both sides of the bar model like that.

And so is the statement going to be sometimes, always or never true? Well, in this case, it's clearly going to be always true, because if p is greater than q plus two, and if I add a five to both sides, the inequality is going to hold, and let's just represent this using algebra.

So I'll write my original inequality above this.

p is greater than q plus two.

And what I've done here is, I have just added five to both sides.

So I get p plus five is greater than q plus seven.

Okay.

Have a think about the next one.

2p is greater than 2q plus four.

Okay.

So once again, I can draw the same bar model, to represent my starting inequality, which is p is greater than q plus two.

And now if I've got 2p, well that is just adding another p, and if I've got 2q plus four, well this is the same thing as two lots of q plus two.

It's the same thing, because if I expand that bracket, that gives me 2q plus four.

So again, all I'm doing is I am doubling both of my starting bars.

So it's going to look like this.

So I've gone from p to 2p and I've gone from a q and two, 2q and two twos or 2q plus four.

So again, is this statement going to be, is this inequality going to be sometimes, always or never true? Well, again, it's going to be always true.

And once again, showing this with algebra, I'll write the original inequality on the top, p is greater than q plus two.

What have I done to both sides? Well, I have multiplied by two.

So once again, the message is that just like I've balanced equations, if I do the same thing to both sides on the equality, that inequality will still hold for the other inequality.

Okay.

In the independent task, you're going to explore this a bit yourself.

Okay So very similar to the previous connect task, so here am got an inequality, a very simple one, A is greater than B.

So you've got six inequalities to look at, and you have to decide, whether they are always, sometimes or never true.

It may be worth drawing out an inequality, a bar model to help you out.

Or you may just want to have a think about this, using your sort of mathematical sense.

So pause the video now and just spend three or four minutes, having a think about these six inequalities.

Okay.

On the next slide, I'm going to show you the answers.

Okay.

So here are the answers then, and let's go through these very quickly.

So the first one is always going to be true, because if A is bigger than B then adding two to both sides, won't change that A plus two will still be bigger than B plus two.

The one below that is also always true, because again, I'm just multiplying both sides of the inequality by two.

So it will still be true.

The third one, the one that is sometimes true is an interesting one.

And this really depends on how big A and B are, because if five is quite small, let's say it just goes up to here.

Then A will still be bigger than B plus five.

But if the five is at like this, then B plus five will be bigger.

So it depends on how big A and B are.

The one below that is going to be always true because this time I'm taking one away from B, so it is going to be smaller than A.

The fifth one is going to be never true.

And the final one, minus A and comparing minus A and minus B.

Well, it's quite hard to see these negatives with the bar models, so I can have a think about different values here.

So let's say for example, that A is equal to 10, and B has to be something smaller, so let's say it's eight, then let's compare minus A and minus B.

So minus A is minus 10 and minus B is minus eight.

So, which has bigger? Well, again, thinking about this on a number line, minus 10 is going to be to the left of minus eight.

So A is going to be smaller than minus eight.

So at the final statement, is going to be always true.

All right.

So again, in the explore task, we are thinking about bar models here, to help us understand inequalities.

So again, I've got some different colours that represent different letters and numbers.

And this time, you can imagine this pattern repeating in the future, because that is going to help us with some of these questions.

So, you have got four pairs of cards, of statements, and you need to decide, whether to put an equals sign, a less than or greater than in between each pair of cards.

So let's do the first one together.

I've got 3q and 4r.

So let's have a look at the diagram, 3q I can see that the q is this a light blue one, so three of those is one, two and three and 4r, well, that's the green stripe one, one, two, three, four.

And I noticed that they are the same length, so it's going to be an equals sign in here.

Okay.

So I'll leave you to think about the rest, have think for a couple of minutes.

Pause the video now, to have a think about the rest of these questions.

Okay, great.

So the one below that, well, this is again thinking about what we have learnt in this lesson, which is that, to get from 3q to 12q, I'm going to have to multiply it by four.

I can't actually see this in the diagram.

So I have to imagine multiplying it by four.

But if I multiply a four by four, I'm going to get 16r.

So 12q would be the same as 16r.

But in the case that these, given that these are both positive numbers, then 12q is going to be a greater than 15r.

And the next one, 2p plus 4.

5 is going to be this bit here.

And I'm comparing that to 3q, which is the whole of this block.

So 3q is bigger, so I can write that inequality.

And the second one is a little bit tricky, but you could have noticed that if I said, 2p plus nine, that would take me up to the same length as 3q.

So that will be equal to 3q.

And then if I'm multiplied both of these by five, I would keep the equality here because a 2p plus nine, all times by five gives me 10p plus 45, and also 3q times by five gives me 15q as well.

So this is going to be true.

Okay.

So that is it for this lesson.

And in the final lesson on inequalities, we're going to be continuing with this theme of having a look at the relationship between different inequalities and equalities as well.

I hope you've enjoyed this lesson.

Thanks very much for watching and see you next time.

Bye, bye.