Lesson video

Welcome and well done for making the decision to learn using this video today.

My name is Ms. Davis, and I'm gonna be helping you as you work your way through this lesson.

There's lots of opportunities for you to really think today and test your understanding.

Make sure you are pausing the video to give yourself that chance to think.

And then I'll help you using any hints and tips as we go through.

If you've got everything you need, then let's get started.

Welcome to this lesson on linear inequalities in context.

We're gonna bring all our inequality skills together today and we're gonna look at applying those to real life situations.

If you'd like a reminder as to what an inequality is, pause the video and have a read through that now.

So we're gonna start by writing inequalities from real life contexts.

Now, inequalities are really useful.

They are common in real life situations.

They are used whenever there's some kind of constraint, so you've probably used inequalities loads already before you studied it as a branch of mathematics.

We use a range of phrases in English to describe inequalities, and it's really important that we are accurate with these and people are not always accurate with the language they use when describing inequalities.

Sofia says, "You have to be over 18 to see a certain film at the cinema." Laura says, "Actually you have to be at least 18." What's the difference between these? Right, well the phrase over 18 implies greater than 18.

So we can write the inequality A is greater than 18, whereas the phrase at least implies greater than or equal to 18.

So we could write this inequality.

If there's an age restriction on the film, that means you have to be at least that age.

So if it's a 15, you can be 15 or over.

Now the differences between these will also depend if we're referring to age in years, or as a continuous measure of time.

So if A is greater than 18, it means you have to be 19, we know that's wrong.

But if that means you just have to be over 18 to the minute, or to the second, then that's less of an issue.

So greater than or equal to, and greater than 18 are gonna be very similar if we're talking about age as a continuous measure.

By using inequalities, we can be much more precise with our meaning.

Sofia says, "The flight information for my ski trip says my suitcase can be no more than 20 kilogrammes." How can we write that as an inequality? Right, we can use any variable we like.

So A is less than or equal to 20, where A is mass in kilogrammes.

The phrase 'No more than' implies that 20 kilogrammes exactly is allowed.

So that's why I've used the less than or equal to symbol.

Just like when you form equations, it's important that you define your variables and include units, which is why I've said A is mass in kilogrammes.

Laura says, "A climbing centre says you need to be at least 10 kilogrammes, but under 150 kilogrammes to climb." How could we write that as an inequality? We could write it like this, and again, let's define our variable.

You can use any variable you like, I've used M, and I've said where it's mass in kilogrammes.

Now under 150 kilogrammes implies that 150 kilogrammes exactly is not allowed.

Now we don't know if that's what the climbing centre means, but they've used that language of under 150 kilogrammes.

So if we're being precise with language, that means 150 kilogrammes would not be allowed.

At least 10 kilogrammes means 10 kilogrammes is fine.

Sometimes in real life we need to understand the context to be clear about restrictions.

A sign at a theme park reads, "You must be 1.

2 metres to ride." Laura says, "I assume it doesn't mean you have to be exactly 1.

2 metres to ride.

That's ridiculous." What do you think this actually means, and write it as an inequality.

Right, it probably means at least 1.

2 metres.

It's not obvious though with its wording, you have to understand the idea that a theme park, you have to be taller than certain heights to ride.

But there might be other rides where you have to be less than a certain height to ride, so this is really unclear.

That is why maths is fantastic because it is really obvious what we mean when we use an inequality symbol.

We could write H is greater than or equal to 1.

2 where H is height in metres, and there's no room to misinterpret that.

There's probably an upper height limit as well.

There's probably a height where you'd be too tall to ride, but because there's a limit on how tall humans grow, it's probably not necessary to state that on the ride.

Right, your turn.

I'd like you to match up the sentences to the inequalities.

Let's have a look.

At most 18 means less than or equal to 18.

No less than 18 means greater than or equal to 18.

Over 18 is X is greater than 18.

Less than 18 would be X is less than 18.

That's an easy one.

I'm hoping you can see now why the mathematics is really clear where sometimes the language is difficult to interpret.

Right, Jacob's cousin has just applied for a provisional driving licence.

How old could they be? Do you know this? How old do you need to be? At the current time you can apply for a provisional licence at 15 years and nine months, and that's because you can ride some motorbikes as a learner at 16, and drive a car as a learner at 17.

So that's why you can actually apply for a provisional licence at 15 years and nine months.

Let's see if we can convert that into something easier to use.

So nine months was 12 months in a year, so 9/12 is the same as 3/4 or 0.

75.

So 15 years and nine months we could write as 15.

75.

As an inequality then, A could be greater than or equal to 15.

75, where A is their age in years.

Jacob says "Actually they're 17 and studying for their theory test." I wonder if you've come across any of these road signs before? Have a look at these.

Do you know what any of them mean? This might be something you're interested in looking out for next time you're on a car journey.

Let's have a look.

This one means "No driving over 20 miles per hour." The second one, "No vehicles over 9.

9 metres long." This third one, "No vehicles over 4.

6 metres high," and often they have the height in feet and inches like this one as well.

And this last one, "No vehicles over 7.

5 tonnes." Now again, you'd have to understand the context and the English road signs to be able to interpret these.

The red circle means a prohibition, so it's prohibited from driving over 20 miles per hour.

It's prohibited for a vehicle to be over 9.

9 metres long on that section of road.

So you do have to understand that in order to know which way the inequality's going to go.

So let's look at writing these as an inequality.

So you can have S has to be less than or equal to 20, where S is the speed in miles per hour.

And the same for our length, height and mass.

Jacob says, "Speed, length and mass cannot be negative, so the range of values could be written like this instead." You can see that he's now entered a lower constraint, 'cause it's impossible for speed to be less than zero.

Now the context tells us the lower constraints, so it's not necessarily important to write it because we know we cannot have a negative speed, length, height or mass.

However, if we wanted to express the range of speeds, then we would need that lower constraint, because we're trying to show all values that the speed could take.

Right, quick check.

I'd like you to match up the sentences to the inequalities.

Off you go.

Positive numbers are greater than zero.

Zero does not count as a positive or a negative.

So negative numbers are less than zero.

All numbers except negatives, you can have N is greater than or equal to zero.

Numbers which are not positive, N is less than or equal to zero.

In some context, the inequalities describe a situation where you can fit one category or another.

So you can have a concession ticket for a park if you are 10 years and under, or 60 and over.

So if we draw that on a number line, negative values for age are impossible, so the lower constraint is zero.

We've got people aged between zero and 10 inclusive, and then people 60 and over.

Now clearly it's not possible to satisfy both of these inequalities simultaneously.

We can write them with an 'or' statement.

So what we mean here is X is greater than or equal to zero, less than or equal to 10 or X is greater than or equal to 60.

And we can form inequalities from more complicated scenarios as well.

Pause the video and read through that information.

Right, so we're looking at a budgeting problem, we're trying to work out how much we can spend on a food shop.

So it's important that we're working either monthly or yearly, not a mixture of the two.

So I've decided to work monthly.

So I've done 24,000 ÷ 12, and that's gonna get our monthly salary.

Net salary means your take home pay, so once you've sorted out paying tax and pensions, and student loans and that sort of thing.

So we're gonna add up all the other things we need to pay for.

So bills, mortgages, savings, and we get 1,662.

So our food shop, we're gonna do four food shops a month.

So four lots of our food shop.

So 4x + 1,662 has to be less than or equal to 2000.

We don't need to spend our whole 2000, it's not a challenge, we just need to make sure that's under or equal to £2000.

And again, let's define what X is.

So X is the cost of a food shop in Pounds.

Sometimes we might need to use some of our shape knowledge.

So the width of a rectangular flower bed needs to be two metres.

You have 18 metres of border material.

Assuming the border is placed around all four sides, write an inequality for the possible lengths.

Well, a diagram's definitely gonna help us.

There's our rectangular flower bed.

It needs to be two metres wide, but it can be any length.

So we can form the inequality, 2x + 4 is less than or equal to 18.

We don't need to use up all our border material, we just need to make sure we have enough.

In this context, length must be positive.

Zero length would not form a rectangle.

So we can write two inequalities.

The 2x + 4 is less than or equal to 18, and X is greater than zero.

We can't combine them and write them as one inequality at the moment, because one inequality is about 2x + 4, and the other one's just about X, so we can just leave them separate.

Right, your turn.

The optimal viewing distance for a TV is 1.

2 times the screen size.

I dunno if you knew that.

Aisha's living room allows her to put the sofa a maximum of 2.

1 metres from the TV.

She writes this inequality, 1.

2x is less than or equal to 210.

I suggest you read the information again.

What does X represent in this context? Right, so if you times the screen size by 1.

2, that gets you the distance from the TV.

So she's saying she can put her sofa as far back as 2.

1 metres, or 210 centimetres.

So that's why 1.

2 times the screen size has to be less than or equal to 210.

That means X is the size of the TV she can have in centimetres.

So Aisha wants a TV which is at least 50 inches, assuming one inch is 2.

54 centimetres, she writes this inequality, can you spot what is wrong with the inequality she has formed? Right, well done if you spotted this, it was quite difficult.

It's the X that has to be greater than or equal to 127, not the 1.

2x.

So actually she has to write these as two separate inequalities.

1.

2x is less than or equal to 210, 'cause she can only get her sofa 210 centimetres back, but X has to be greater than or equal to 127, 'cause she wants her screen size to be larger or equal to 50 inches.

And of course we need X to be defined as distance in centimetres.

Right, your turn.

I'd like you to write an inequality to show the range of possible values for each description.

Off you go.

And then you've got three there to do for question two, because these are in context, you need to define your variables.

And then you've got two more to do there.

For question three, Alex this time wants to work out what size of TV he can have.

So read through his constraints, can you write an inequality? And for four, we've got another budgeting problem.

Any money not spent on the things mentioned could then be spent on holidays.

Right, have a go at these two.

Off you go.

Let's have a look at our answers.

I'd like you to pause video, and check through these.

Just a reminder that zero is not a positive number, so if you want positive numbers, you need your value to be greater than zero.

For C, you just need to write N squared is less than nine for now.

If you try to work out what that means N must be, then N has to be greater than -3, but less than three.

But you don't need to worry about that for the moment, we're just forming these inequalities.

Check the rest of your answers.

So for A, where A is greater than or equal to 13.

You can use any variable you like as long as you define it as age in years.

S is gotta be greater than or equal to 30, but less than or equal to 50, where S is speed in miles per hour.

For C, A has gotta be greater than or equal to two, but less than 10 where A is age in years.

For D, we've got A is greater than or equal to 21, but less than 70.

And for E, we need an 'or' statement, because A is less than five, or A is greater than or equal to 75.

If you said that age cannot be negative, you may have written that as A is greater than or equal to zero, less than five, or A is greater than or equal to 75.

Your answer to this one may depend if you worked in centimetres or metres.

Working in centimetres, you'd have 1.

2x is less than or equal to 200, and X is greater than or equal to 107, where X is the screen size in centimetres.

If you try to combine those as a single inequality, I've written that answer underneath as well.

If you've done it in metres, that would be perfectly acceptable as well.

your answers would just be a hundred times smaller.

For four, again, it depends if you're working yearly or monthly.

This one looked easier to do yearly, so I've added up all the constraints yearly, which meant I had to do 200 X 12, and 300 X 12 to work out the food and the transport per year.

And if I add those all up, you get 25,626.

Then I can form the inequality, 25,626 + X has gotta be less than or equal to 28,000.

If you started to solve that, or you wrote an equivalent inequality, that's fine as well.

And there we've got our rectangles.

So for the first one, you get 6x is less than 100, but width needs to be positive.

So 6x is less than 100, and X is greater than zero.

For six, you've got 2x + 17 is less than or equal to 45, and X is greater than zero.

And again, you need to define X as width in centimetres.

Well done.

You've done all the hard work with forming inequalities now we're just gonna have a look at solving these.

Once we've formed an inequality, we can solve it to find the range of possible solutions.

So let's look at Aisha's TV again.

We formed the previous inequality that 1.

2x has to be less than or equal to 210, and X is greater than or equal to 127.

So all we need to do now is solve that.

So if we divide both sides by 1.

2, we get X is less than or equal to 175.

And don't forget, she also needs it to be bigger than 50 inches, which is 127 centimetres.

So we get X is greater than or equal to 127, less than or equal to 175.

Let's look at our budgeting problem.

So there's the key piece of information we had before.

Feel free to read through that again if you need to.

And we wrote this inequality.

So this was the food shops one's we're doing, four food shops a month, and we want to know how much we can spend on each food shop.

So four lots of the cost of the food shop, plus 1,662 has to be less than or equal to 2000.

So again, if we solve that, we can subtract 1,662 from both sides, and divide by four.

So we get X is less than or equal to 84.

50.

So that's £84.

50.

In context, we know this cannot be a negative number.

You can't spend a negative amount of Pounds at a food shop.

So we could write that as, X is greater than or equal to zero, less than or to £84.

50.

Right, quick check.

We formed these inequalities for the dimensions of our flower bed.

What is the range of possible values for the length X? Give this one a go.

So, solving our first inequality, you can subtract four, and divide by two.

So your X is less than or equal to seven.

We know that X has to be positive, so X is greater than zero, but less than or equal to seven where X is the length of the flower bed in metres.

Once we have a solution we can check using substitution.

So let's check that top constraint.

Let's see what happens when X is seven.

You get two lots of 7 + 4 is less than or equal to 18.

So you get 18 is less than or equal to 18.

So now let's check a value smaller than seven to check if that also works.

So let's try six.

And you get 16 is less than or equal to 18, so that's fine.

And any value less than seven will work as well, except it needs to be positive because it's a length.

We can form and solve inequalities from other number puzzles.

Sofia says, "Laura and I are starting with the same number.

I add three then multiply it by two.

Laura multiplies by four, then subtracts five." Her new number is smaller than Sofia's.

How could we form an inequality for this information? What do you think? Well, for Sofia she's adding three, then multiplying by two.

So if I use brackets, that's two lots of X + 3.

Laura's number was smaller, so Sofia's number has to be greater than, 4x - 5, which is Laura's number.

To work out what number they could have started on, we just need to solve this.

So I've expanded the bracket, so I've got 2x + 6 is greater than 4x - 5, Then I can subtract 2x, add five, and a divide by two.

So X has to be less than 5.

5.

There are often times in mathematics where we want to know what values of the variable make the expression positive.

So for what values of X is the expression 5 - 2x positive? Well, firstly we can set up an inequality.

Zero is not a positive number, so 5 - 2x has to be greater than zero.

Now there are options to solving this.

Personally I like to avoid having to divide by a negative and change my inequality sign, so I'm gonna start by adding 2x.

Dividing by two gives me X is less than 2.

5.

So this expression is positive whenever X is less than 2.

5.

Sofia says, "When we want to evaluate a square root of an expression, it cannot be negative." So for what values of X can we evaluate this? How would you work this one out? Well, the expression 5x - 8 cannot be negative, so it must be zero or greater, and then we can solve that inequality.

5x is greater than or equal to eight, X is greater than or equal to eight over five, or 1.

6.

So the values of X that we can evaluate that square root for is when X is greater than or equal to 1.

6.

For what values of X is the expression 10 - 5x negative.

Well done if you said, X is greater than two, let's look at why.

So 10 - 5x has gotta be less than zero to be negative.

So 5x has gotta be greater than 10, so X has gotta be greater than two.

Right, time to put that all into practise.

So for question one, what values of X are these expressions positive? And for question two you've got the same problem as in part one, but this time I want you to solve that inequality.

How much could they spend on holidays a year? Give those ones a go.

And again, these are the same problems you looked at in task one, but I would like a solution this time.

It's a new problem this time.

So the length of a different rectangle needs to be exactly five centimetres longer than the width.

The perimeter is greater than 16 centimetres, but no more than 30 centimetres.

So you're gonna need to set yourself up an inequality, drawing a diagram might help, and then solve that.

Give that one a go.

Sofia and Laura are starting with the same number.

Sofia halves hers, then subtracts six.

Laura doubles hers, then subtracts 10.

Sofia says, "I finish with a negative number greater than the number Laura finishes with." Can you set yourself up an inequality then solve it to find the range of values they could have started with, then I'd like you to test a value to check your solution.

Come back when you're ready for the next bit.

Question seven, to form a triangle, the two smaller size must sum to a value larger than the longest side.

And that's true.

The expressions for the sides are shown below.

The base of 10 is the longest side.

If the perimeter is than 50 centimetres, what could the length X be? So you've got two things to think about there.

The fact that the two shorter sides have to add to a value larger than 10, and the perimeter is less than 50.

When you're happy with your answers, we'll look through those together.

Let's have a look.

So for one, X has to be greater than -0.

75, or negative 3/4.

For B, X has to be less than two.

And for C, X has gotta be greater than 0.

5, or greater than a half.

For question two, we just need to solve the inequality we had earlier.

You get X is less than or equal to 2,374.

Money cannot be negative, so X has gotta be greater than or equal to zero, less than or equal to 2,374.

And we need to define X, X is yearly amounts spent on holidays in Pounds.

Then we had our rectangles from part one.

6x has gotta be less than 100.

So X has gotta be less than 50 over three.

But X also has to be positive.

So X has gotta be greater than zero.

For question four, if we solve that inequality, X has to be less than or equal to 14, but X is also positive, so X is greater than zero, less than or equal to 14.

And again X is width in centimetres.

For question five, you need to draw your diagram, and set up your inequality.

So 4x + 10 has to be greater than 16, but less than or equal to 30, 'cause it says no more than 30.

Solving that you get 4x is greater than six, less than or equal to 20, X is greater than 1.

5, less than or equal to five, where X is width centimetres.

Question six, if we're gonna set up an inequality X over two subtract six has to be greater than 2x - 10, but less than zero 'cause Sofia's number was negative.

If we solve the left hand side, you get, X is less than eight over three.

Feel free to pause video, and check each step of my working.

If you solve the right hand side, we get X is less than 12.

So the solution for both will be, when X is less than eight over three.

And then to check a solution, where you've got many options to choose from, I've gone with when X is two.

So if you half and subtract six, you get -5.

If you double and subtract 10, you'd get -6, which means Sofia's numbers will be negative, and greater than Laura's.

And finally 3x + 1 has to be greater than 10, 'cause remember the two shorter sides has to add to a value larger than the longest side.

But also the whole perimeter has to be less than 50.

So 3x + 11 is less than 50.

If you solve those separately, you get X is greater than three, but less than 13.

Again, if you want to pause video and check through my working, feel free to do that now.

Fantastic.

Hopefully you've seen today that inequalities are really common in real life.

They are used when a constraint is needed, and often we use language, we have to be really accurate with the language we're using, 'cause the type of words we use will depend on what sort of inequality we are referring to.

And then we've looked at solving inequalities in context as well.

Thank you for joining me today.

You've worked incredibly hard, and I hope to see you back for some more lessons in the future.