Lesson video

Hello everyone it's Mr. Miller here welcome to the fourth lesson on inequalities.

And in this lesson we're going to be looking at solving inequalities.

Okay, so firstly I hope you're doing well let's have a look at the 'Try this' task to start.

So here I've got an inequality 13 is greater than or equal to 4 plus 3 x.

And what I want you to think about is what different integer values or the whole number values that could x have.

So, for example if you wanted to try, um.

So if x was equal to one then let's see what we have.

We would have 13 is greater than or equal to four plus three x and if x is one then three x is simply three.

So, uh what's this four plus three is equal to seven, so does this work? Yes it does, because thirteen is greater than or equal to seven.

Uh, so x equal one does work.

So what I want you to do is I want you to try different values of x and see what works here.

Okay so hope that you tried some other values of x and you found some interesting things.

Uh, so if you tried x equals two you would've found that this works as well because you'd have 13 is greater than or equal to four plus six.

Uh, so that works.

Um, if you had x equals three then that also works because then you'd have thirteen is greater than or equal to four plus nine, which is thirteen.

So, uh, that works as well.

But if you tried a number higher than three such as four then you should've found that this doesn't work because then you'd have 13 is greater than or equal to four plus twelve.

Which is 16.

So, uh that's doesn't work either.

And you could've also tried numbers that are higher than four but you should've found that will never work.

Um, and also negative numbers as well work as well.

Now, let's put all of this on a number line.

Uh, which you can see there.

And if I label this number line, so starting off with four at the top and then going down, what you would find is that when we tried one, two, or three that worked.

The inequality did hold.

But anything less that one would also work so anything down here would've worked.

Um, but the moment we past three it doesn't work.

Uh, so anything higher than three doesn't work so what we could say, if I look at this number line is that anything which is three or lower does make this inequality hold.

Anything higher than three and we couldn't make this inequality true.

So anyway this is a interesting result and we're going to look at this in more detail in a connect task now.

Okay so here is a connect task and what I want you to think about is what's the same and what's different with these two things in the green boxes.

Have a moment to think about what's the same and what's different here.

Okay, so um as you probably noticed that this is the what's on the right is exactly what we had before in the try this.

And what is similar of course is that the first one has um the same numbers.

We got thirteen, we got four, we got a plus three x and of course what's different is that the first one has an equals sign and the second one has an inequality.

Now you've done lots of solving equations before so how would you solve the equation on the left hand side? Well you would do some kind of balancing method so start off by subtracting four from both sides so you have nine equals three x.

And then to find out what x is uh you would divide both sides by three so you have three equals x or just x equals three.

And that would be the answer.

Now to solve the inequality it turns out that you would do exactly the same steps that you would do if it were an equation.

Uh, so I'm going to do exactly the same thing.

Uh, my first step is I'm going to subtract four from both sides and I have nine which is greater than or equal to three x.

And then my second step just as before I divide both sides by three so I have three is greater than or equal to x.

Or if I just turn around the inequality I can say that x is less than or equal to three.

So a couple of important things to point out here which is that obviously the steps that you do are the same and-- but you must make sure that when you're doing an equation you keep the equals sign.

But if you're doing an inequality you keep the inequality sign.

So an important difference.

Now, if I were to show you the number line that we plotted earlier.

Uh, so what we found out is that anything which was three or less would satisfy the inequality.

What do you notice about the equation? Well, the only value of x which satisfies the equation is three so we can say that the inequality has a lot more of different values of x that would satisfy it.

In fact there are infinite values of x that satisfy it.

So the main take away from this slide and what we're going to do more practises on the independent task is that when we have an inequality to solve, we solve it in just the same way that we solve the equation.

Let's have a look at some more examples.

Okay, so here is the independent task there are three different inequalities for you to solve.

So pause the video now to copy down these three examples and have a go at solving these inequalities.

Remember your solving these inequalities imagining that there was an equals sign instead of an inequality but you're still going to do the same thing as if you were solving an equation.

Pause the video now to have a go at these three examples.

Great and now we're going to go through them so if you haven't had a go make sure that you do that.

And in the first example the first thing that we're going to do is subtract seven from both sides.

So I have four x is less than or equal to twenty.

And then I'm going to divide by four so I have x is less than or equal to five and that is my answer.

The next one, the first thing I'm going to do is subtract a half from both sides so I have a half is greater than x over 10.

And then what I'm going to do is I'm going to multiply by 10.

And a half times by ten is equal to five.

So I have five is greater than x.

Or I can write it as x is less than five.

The final one well I've got x on both sides so I need to find the smaller coefficient of x and get rid of that so subtract x from both sides and I get eight is less than two x minus 10 I'm then going to add ten to both sides so I have 18 is less than two x.

And then finally divide by two so I get nine is less than x.

Or x is greater than nine.

So make sure you have all these answers down into your notes.

Um, and make sure that you've noticed that as I've been doing these I keep the inequality the same throughout all of these examples.

Okay let's move on to the explore task when you're ready.

Okay, so here's the explore task let's have a read.

So, if a is equal to five, how many different ways can you use the expression cards to complete the inequality frame.

And what if it's 10 what if it's 100 worry about those later, but for now what you need to do is imagine that a is equal to five how many ways can you fit the frames into this inequality.

Okay, and if you're a little bit stuck the first thing that you should do is you should find out what each of these expressions is equal to with a is equal to five.

So two a minus five is going to be two times by five which is ten minus five.

Which is equal to five.

The next one is going to be five minus five which is zero etc.

Once you've done that then you can find different ways to put this into the inequality.

So pause the video now and make sure that you have a good go at this explore task.

Okay brilliant so if you are count the remaining ones at ten plus a it's going to be 15 and a plus five is going to be 10.

Uh, so you want to rearrange these into the inequality sign.

So you could have first of all the number way to do this you could have 10 plus a which is 15.

Um, and then you could have a plus five in here which is 10.

So those two would sum up to 25 and then you put the other two in there two a minus five and five minus a which is zero.

So that will work because 25 which is the combined total of the two on the left hand side is bigger than five.

There's one more way to do that which is to actually swap around the a plus five and the two a minus five.

So if I show you that very, very quickly.

Yup, so all you need to do is have the a plus five here and the two a minus five here.

And that is-- those are the only two ways to complete it with a is equal to five.

But if a is equal to 10 or if a equals 100 if you work those out you should've found that they're actually three different ways to complete the inequality frame.

Uh, compared to the two different ways when a is equal to five.

And that's really interesting and I'd encourage you to have a think about why that is.

And one way that, um, you could think about it is if I just rub these out very quickly.

One way for you to think about this is well imagine that you had two a minus five in here and you had a plus five in here.

And then you had the other two here so five minus a and ten plus a.

Well if you just added these together, without worrying about the value of a first of all well if you collected like-terms now you'd have two a minus five plus a plus five.

And the fives cancel out so you get three a here and the right hand side the A's cancel out so you get 15.

So this gives you three A is greater than 15.

Um, which goes down to A is greater than five.

So three A is greater than 15, A is greater than five.

So what this says is this will only work as long as a is greater than five.

So it doesn't work if a equals five but if a equals 10 or equal 100 that you could have a card in this way.

So yeah that is it for this lesson hope you've enjoyed this lesson and it's an important one because in the following lessons we're going to be looking at solving inequalities in more detail.

So, um, make sure that you are ready for that and fully understood how to solve an inequality.

Thanks very much for watching and I'll see you next time.

Bye bye.