Lesson video

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

My name is Ms. Davies 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 inequalities on a number line.

By the end of the lesson, you'll be able to represent inequalities on a number line and what values satisfy an inequality based on the notation used.

So an inequality is used to show that one expression may not be equal to another.

You've probably used these loads in mathematics already.

We're really gonna delve into what they mean today.

So let's start by looking at representing a single inequality on a number line.

Andeep has written this inequality.

We read it as X is less than 4.

Jacob said, "That means X can be 0, 1, 2, or 3." Aisha says, "It could also be any negative number, remember." And Andeep says, "This includes non-integers less than 4 as well, such as 3.

9, 1.

47 or 7 over 3." In fact, this includes an infinite number of values which are less than 4 and that's why that inequality notation is really useful.

We're not gonna want to write all values that satisfy that inequality.

It's not gonna be possible.

So using that notation X is less than 4 is really helpful.

Another way to do this will be to use a number line.

So we can show this inequality in this way.

So I've drawn a number line, I've put a circle above 4 and you'll see I've got an arrow going left.

Let's look at these key features in more detail.

Now this circle is an open circle.

That means it is not shaded in.

It's above 4 'cause 4 is not included in our value, but it's the boundary.

Any value up to 4 is included.

Then we have an arrow pointing left to tell us that all values less than 4 are included.

Without that arrow, we wouldn't know which side of 4 we were looking and we wouldn't know that we're allowed to include any value to the left of 4.

Now it does not matter how big your number line is as long as it includes the value 4, because we need to draw the circle above 4.

It also does not matter how long we draw the arrow.

It doesn't matter that it only goes to negative 1.

It could be shorter, it could be longer.

The fact that we've put the arrow on the end says that that's gonna include every value infinitely to the left of 4.

Jacob says, "I want to change the inequality to X is less than or equal to 4." That symbol there, I'm sure you've seen before, reads as X is less than or equal to 4.

So what do you think that's gonna look like on a number line? I'll show you.

This includes the number 4 itself this time.

So what we do is we shade in the circle above 4 to show that we are including 4.

We call that a closed circle.

Again, we want the arrow pointing left to say that we are looking at values less than or equal to 4, just any value to the left of 4 and including 4 'cause that circle is shaded.

So those are the two inequalities we've looked at so far.

What inequality do you think has been drawn on this number line? Pause the video.

What do you reckon? Well done if you said X is greater than 4.

Doesn't matter what variable you use here because we're not representing anything in particular at the moment.

So X is greater than 4.

Make sure you haven't got the greater than or equal to sign by mistake.

We are not including 4.

That circle is not shaded in.

Right, quick check.

I'd like you to match the inequalities to the number lines.

Off you go.

Let's have a look.

X is greater than 2, open circle at 2, arrow to the right.

X is less than 2, open circle at 2, arrow to the left.

X is greater than or equal to 2, closed circle at 2, arrow to the right.

And X is less than or equal to 2, closed circle at 2, arrow to the left.

Which inequality has been drawn on this number line? See if you can write it using the correct notation.

Well done if you said X is greater than or equal to negative 3.

Your turn to draw one.

What would this inequality look like on a number line? Sometimes reading it out loud can help.

So this is X is less than negative 1.

So you need an open circle at negative 1 and an arrow to the left.

Aisha says, "I'm thinking of a number greater than negative 3 but less than 2." She's got two restrictions here.

What number could she be thinking of? Give me some suggestions.

Right, there's loads of suggestions that you could have.

There's some of the ones I've gone for.

Obviously if you are thinking of integers, you could have had negative 2, negative 1, 0 or 1.

You can't have 2 'cause she said less than 2.

So 2 is not included.

And of course there's a whole range of non-integer values as well you could have chosen.

Aisha said that her number is greater than negative 3.

So that could not be negative 3 itself.

And like we've just said, less than 2 does not include 2.

Right, you've possibly seen this before.

We can write this as a single inequality.

Now when we read this, we don't necessarily read it from left to right anymore.

We can start with what X is and how that relates to the values.

So we say that X is greater than negative 3 but less than 2.

And that's why I'm reading it that way because I'm defining what X is even though the negative 3 is on the left.

There's nothing wrong with saying negative 3 is less than X, which is less than 2.

But it makes more sense with our question to define that X is greater than negative 3 but less than 2.

So this can be represented on a number line using two circles for the two restrictions.

So I've got circle at negative 3 and a circle at 2.

They are open 'cause we're not including those values and any value between them is included, so I connect them with a line and that's how we show any value greater than negative 3, but less than 2.

Andeep has drawn this inequality.

Which inequality is Andeep representing? How can we write this? Lovely.

We've got X is greater than negative 1 and less than or equal to 4.

Notice that we start with the smaller value on the left when we're writing this, just like on a number line.

The negative 1 is less than X, which is less than or equal to 4.

Again, any variable can be used here.

I'm just using X for ease at the moment.

Your turn.

I'd like you to match the inequalities to the number lines.

Off you go.

Lovely.

A, we've got X is greater than 0 and less than 3.

We need an open circle at 0 and 3 and a line connecting them.

For B, X is greater than or equal to 0 but less than 3.

So we need a closed circle at 0 and an open circle at 3 and a line connecting them.

X is greater than 3 but less than or equal to 3.

We need an open circle at 0, closed circle at 3 and a line connecting them.

And finally D is gonna match up that second one down.

We've got closed circle at 0, closed circle at 3 'cause we want values greater than or equal to 0 and less than or equal to 3.

Don't forget your line connecting them.

Jacob says, "We were investigating squares recently.

We found that values less than 0, squaring them made them bigger.

Negative values, for example, when you square them, get bigger.

The same is true for values greater than one." Why can Jacob not use that inequality notation to define numbers that produce larger values when squared? Right, well there's no values that are simultaneously greater than 1 and less than 0.

He's written X is greater than 1 but less than 0.

Well that's impossible.

Instead we need to write this as two separate inequalities.

So numbers of the form X is less than 0 or X is greater than 1 produce larger values when squared.

Let's look at what this looks like on a number line.

So we write them as two separate inequalities so we can draw them as two separate inequalities.

So X is less than 0 and X is greater than 1.

We've got two separate inequalities and we're happy with a value in either region.

The only values that are not included are values between 0 and 1 inclusive.

Quick check.

Can you match the inequalities to the number lines this time? Let's have a look.

So A, we want values greater than 1 but less than or equal to 3.

So that's that second one.

For B, we've got an or so we're probably gonna have two different arrows.

We've got X is less than 1 or X is greater than or equal to 3.

That's the top one.

For C, we've got another or statement, X is less than or equal to 1 or X is greater than 3.

That's that bottom one.

For D, X is greater than or equal to 1 and less than 3.

So it's that third one down.

Time for practise.

I'd like to explain what mistake has been made when each inequality has been drawn on a number line.

Try those four.

Well done.

Your turn to draw now.

Can you represent each of these on a number line? Pay attention to C.

I've been a bit mean and I've written it the opposite way around to how we've been writing it.

So think carefully about what that one is telling you before you draw it.

For question 3, have a go at representing these on a number line.

Be careful with D, E, and F.

You'll see I've changed the scale on the number line.

Question 4, I'd like you to write down the inequality represented on each number line.

Give those a go.

And finally, write down the inequality represented on each of these number lines.

Again, be careful with some of the scales.

If you work out what each little line is representing to help you get the right values.

Let's have a look.

So the first one, we haven't got a circle to indicate the constraint.

We want X is less than 4, that's our restriction.

We need a circle above 4.

Be an open circle in this case as well.

For B, we need an arrow 'cause we need to show that all values greater than 2 are included.

We don't just stop at 4.

For C, we've got the wrong kind of circle.

It should be a closed circle to show that 3 is included.

And for D, we should connect those with a line instead of using arrows to show that only values between 1 and 4 are included.

Otherwise we might think that all values less than 4 are included because there's an arrow pointing left of 4.

So using that line to connect the two just shows us we're only looking for values between the two.

For question 2, let's see what these should look like.

Careful with C, that X is greater than or equal to negative 1.

So we need a closed circle at negative 1 and arrow to the right.

Well done if you got that.

Pause the video if you need to check those again.

For question 3, we've got an or statement, we should have two arrows.

Same for B.

For C, we need values between negative 2 and negative 1, inclusive.

Make sure you've filled in those circles.

Got an or statement for D, again, I've been a bit mean and I've written X is greater than or equal to 2 the opposite way round to how we often see it.

For E, and then for F.

For 4, we've got X is less than 3.

For B, X is greater than or equal to 0.

For C, X is greater than or equal to 0 and less than 1.

For D, X is less than or equal to negative 1.

For E, X is greater than 0 and less than 3.

And for F, X is greater than negative 1 and less than or equal to 0.

Again, you might wanna pause and just check you've written those all the correct way round.

And for 5, X is less than or equal to 0 or x is greater than or equal to 2.

For B, X is less than negative 1.

5 or X is greater than 0.

And for C, x is less than negative 0.

2.

Well done if you got that one.

I was a bit mean with some of these scales.

For D, each little line is 4.

So X is greater than 14 but less than or equal to 22.

For E, X is greater than negative 1.

2 but less than 0.

And for F, X is less than or equal to 10 or X is greater than 12.

5.

Again, pause the video if you want to check your answers in more detail.

So now we're gonna look at representing multiple inequalities.

Sometimes when values have to satisfy two inequalities simultaneously, they can be written as a single inequality.

Jacob says, "To go on the 'hopper' at the fair, you have to be at least two years old." "And it's a toddler ride, so you have to be five years or under." How could we write this as a single inequality? Perfect.

Your answer should look like that.

On a number line, we can combine the two inequalities into a single inequality.

So say I drew them as single inequalities, so there's X is greater than or equal to 2 and there is X is less than or equal to 5.

If I want things that satisfy both, I'm looking at that section there between 2 and 5 inclusive so I can draw that overall inequality and that's what we've just seen in the previous part of the lesson.

Sometimes the new inequality is gonna have two constraints or it might just have one.

So let's look at this.

Values that satisfy X is greater than or equal to 2 and x is greater than 4.

Notice that I've written them the opposite way round again.

So I read those by looking for where the X is and then looking at the relationship to the number, which is why I read that as X is greater than or equal to 2 and X is greater than 4.

If I draw those on a number line, Aisha says, "Both the arrows are going the same way this time." Let's see what effect that has.

In order to satisfy both, values must be greater than 4 and all values greater than 4 are also greater than or equal to 2 but they have to be greater than 4 or they don't satisfy that second inequality.

Alright, what about if I want values which satisfy X is less than 5 and X is less than or equal to 3.

They need to satisfy both.

Where am I looking? Let's check on a number line.

If they need to satisfy both, then they need to be less than or equal to 3.

Quick check.

Which inequality shows all values which satisfy X is less than or equal to 4 and x is greater than 1? I've drawn a diagram to help you.

Yeah, X is greater than 1 but less than or equal to 4.

Slightly trickier one.

Which inequality shows all values which satisfy both X is less than or equal to 4 and x is less than 1? Well done if you said that X has to be less than 1.

We don't want values greater than 1 'cause it won't satisfy that inequality X is less than 1.

Let's look at another example.

This time values have to satisfy X is greater than or equal to 2 but less than 6, and also X is greater than 3.

A number line is gonna be really helpful.

So if I draw the first inequality and the second inequality.

Now if I want them to satisfy both, I'm looking at values in that range there, greater than 3 but less than 6.

What integer values are gonna satisfy both inequalities? What do you think? In this case, only 4 and 5.

There'll be lots of other values that satisfy this inequality.

But, if we want integer values, 4 and 5.

Right, what is the range of values which satisfies both the inequalities below? Have a go at this one and then we'll draw the number line to help us.

Okay, let's draw our first inequality and then our second inequality.

To satisfy both, values need to be greater than or equal to 0 but less than 3.

If I use inequality notation, it should look like that and that's what it's gonna look like on our number line.

What integer values satisfy both of these this time? Hopefully you remembered that we're including 0.

So 0, 1 and 2.

Sometimes adding extra constraints does not affect the inequality.

So let's look at values that satisfy all three of these.

On our number line, X has gotta to be greater than or equal to negative 2 but less than 3.

X has gotta be less than or equal to 2.

And that means we can no longer include values greater than 2 and less than 3.

But if I add this third constraint, all our values are already greater than negative 3, so that does not affect the overall inequality.

So to satisfy all three values have to be greater than or equal to negative 2 and less than or equal to 2.

Which diagram shows all values which satisfy both the inequalities shown? Off you go.

Well done if you picked C, values greater than 1, but less than or equal to 5.

Which inequality represents all values which satisfy both of these inequalities? A quick sketch might help.

And we can see this value is greater than 3 but less than or equal to 5.

Time for a practise.

I'd like you to write an inequality for all values which satisfy both the inequalities shown, represent it on a number line, and then select any of the values which satisfy your new inequality.

You've got two to do there.

Question 3, exactly the same idea.

How would you write an overall inequality for all values which satisfy both? Draw it and select any values in your new inequality.

Same again for 4.

Give those two a go.

Question 5, again, I'd like you to write an inequality for all values which satisfy both, represent it on the number line and this time write down all the integer values which satisfy your new inequality.

Then do the same for 6.

And finally, I'd like you to represent both inequalities on the number line and then write an inequality for all values which satisfy both.

Give that one a go.

Let's have a look.

Right, pause the video and check you've written the correct notation and drawn correctly on the number line.

Think carefully about whether they should be open circles or closed circles, whether they're less than signs or less than or equal to signs.

And then check that you've circled the correct values included in your new inequality for both.

When you're happy with your answers, go on to the next one.

For question 3A, we need X is less than or equal to 1.

And you can see how we've drawn that on the number line and outta those answers, that includes negative 1 and 1.

For 4, we've got X is greater than or equal to 1.

You can see that on our number line and that's gonna include six fifths.

We know that's greater than 1 because it's an improper fraction, and 2.

5.

For question 5, again, pause and make sure that you've got the correct inequality notation and the correct diagram.

The integer values are 1 and 2.

For 6, the integer values are negative 1, 0 and 1.

Make sure you're happy with those answers and we'll look at the last one.

Let's have a look.

So for A, we're looking for X to be greater than or equal to negative 4 but less than or equal to 3.

For B, X is greater than or equal to 0, but less than 2.

For C, x is greater than negative 5, but less than 0.

Make sure you haven't written greater than or equal to negative 5.

Our second inequality states that X has to be greater than negative 5.

So that open circle takes priority there.

For D, X is greater than or equal to negative 3 and less than or equal to 1.

Make sure you've spent enough time checking your diagrams and checking your notation.

Fantastic.

I hope you've seen how easy it is to represent inequalities on a number line.

Even though it's easy, it's a really important skill.

It's gonna help you when you are solving inequalities later on.

Really helps you think about what values are included in your answer.

We also know that representing things in different ways can be really useful to being able to use a diagram and use notation can be a really helpful skill.

Thank you for joining us today.

I hope you feel really secure now with your understanding of inequalities.

Remember that sometimes reading them out loud can help you make sure you're getting things the right way round.

If you want to remind yourself of the key things that we've looked at today, pause the video and have a read through those now.

Now you've got your inequalities basics really solidified, I hope that you choose to join us for another lesson in the future and we can see where we can take these skills and develop those even further.

Thank you for joining me.

Have a lovely rest of your day.