Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

Today, we're looking at solution set notation.

This is a lovely bit of maths, which is gonna be useful today as well as being really useful in your future mathematical learning.

So let's find out what it's all about.

Our learning outcome is it'll be able to represent a solution set using set notation.

Some keywords we'll be hearing throughout the lesson.

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

A solution is a value or set of values that can be put in place of an unknown which makes the equation true.

There's two parts to our learning today and we're gonna begin by looking at set notation.

We can write the solution to two different inequalities as one single solution set.

For example, what single inequalities satisfies both of these inequalities x is greater than 3 and x is less than 9.

Considering a number line helps us to see the solution here, there's x is greater than 3, there's x is less than 9 and it's this region we're interested in.

We can express that as one single inequality.

3 is less than x is less than 9.

We mathematicians like efficiency and clarity.

The single solution set is very simple and clear, for this example.

This pair of inequalities, however, poses a problem.

What single inequalities satisfies both of these inequalities? x is less than 3 and x is greater than 9.

Can you see the subtle difference between this example and the one we saw a moment ago? Let's have a look on the number line, x is less than 3 and x is greater than 9.

June says, "This is easy!" 3 is greater than x is greater than 9, "x is less than 3, x is greater than 9." Is June, right? Can we write this as a single solution set like that? Pause and have a think.

Welcome back.

I'm hoping you said, "No we can't write it like that." Let's explore why.

This is not a valid solution.

Because there is no value you can attribute to x that satisfies this inequality.

If we say x is equal to 0, well that's fine, in the case of x is less than 3, but it doesn't work for x is greater than 9, 0 is certainly not greater than 9.

How about x equals 10? Well, it's true for x is greater than 9, but it's not true for x is less than 3, so there's no one value that we can attribute to x that satisfies inequality.

Hence, it's not a valid solution.

There's no single inequality which satisfies both of these inequalities.

Instead, we can say that x is less than 3 or x is greater than 9 and that's the easiest way we can express this, using inequalities, but we could also represent this using set notation.

The set notation for this pair of inequalities looks like this.

When you first look at this, it looks slightly confusing, so let's take a detailed look what each little component means.

It means this set of values in union with this set of values.

We use braces to represent a set.

Informally, you might hear these called curly brackets.

You don't hear me using the phrase curly brackets, you'll hear me calling them braces.

x: means x belongs to the set of values such that, so in that first pair of braces there, x belongs to the set of values such that x is less than 3.

The union symbol tells us that x can be in one set or the other or both.

In this case, we know that the both contains no values because we know there's no value of x, that's in both of those sets.

Quick check.

You've got this so far.

Which set notation describes this pair of inequalities? Pause and have a think.

See if you can spot the right set notation.

Welcome back.

I'm hoping you said it's option B.

Let's have a look at why it was not option A or C.

In the case of option A, we used brackets not braces.

That's not how we write set notation.

We need to use braces.

For C, there was the intersection symbol, not the union symbol.

We know there's no intersection for these two sets.

There's no value whereby x is less than 0 and x is also greater than 2 so there can't be an intersection.

Another check.

Which set notation reflects the inequalities drawn on these number lines? Pause, look at that number line, look at the set notation.

See if you can pick out the right option.

Welcome back.

I'm hoping you said option C, x belongs to the set such that x is less than -4 in union with x belongs to the set such as x is greater than -1.

We can also write this inequality using set notation.

3 is less than x is less than 9 as represented on that number line at the bottom of the screen.

Using set notation, it would look like this.

Well is that not just the same thing? 3 is less than x is less than 9 or x belongs to the set such that 3 is less than x is less than 9.

It might seem at first like our notation has over complicated this example without adding any value, but it is necessary to be able to write this equality as a set.

Let me show you why.

Which values satisfy these inequalities? We can see the inequalities x is less than or equal to 1 and 3 is less than x is less than 9.

We can write the sets of values that satisfy these using set notation.

x belongs to the set such that x is less than or equal to 1, x belongs to the set such that 3 is less than x is less than 9 and if we want to write it all in set notation, then it's that first set in union with that second set.

Quick check.

You can do this.

I'm gonna do one example then ask you to repeat the skill.

Write the set notation to describe these inequalities.

Well I can see the set x belongs to the set such that 0 is less than x is less than or equal to 3, in union with our second set is x is such that x is greater than 5.

Your turn, there's a number line, there's some inequalities on, there's likely to write the set notation to describe them.

Pause and do that now.

Welcome back.

Hopefully you wrote x belongs to the set such that x is less than or equal to 2 in union with x belongs to the set such that 5 is less than or equal to x is less than 8.

Practise time now.

For question one, I'd like to draw inequalities on the number lines to reflect the set notation.

There's 3 examples here for you to try.

Pause and give it a go now.

For question two, we're gonna turn that problem around.

I'd like you to write the set notation having been given these inequalities.

Again, there's 3 examples here.

Have a go at writing the set notation now and don't worry if you make a few errors.

I'll be back with the answers shortly so you can make any corrections necessary.

Welcome back.

Feedback time.

Let's see how we did on question one, we would draw inequalities on the number line to reflect the set notation.

For part A, you should have had two arrows which looked like so.

Be sure to have the solid dot above the 7 to show that x can be equal to 7 and a hollow dot above the 9 to show that x can not be equal to 9, in that case.

For B, we should have had that on our number lines.

Again, be careful about the hollow dots and the solid dots.

For C, our representation should have looked like so.

Question two, we were writing the set notation for these inequalities that we were given.

For A, we should have written that.

For B, our answers should look like that and C was tricky but lovely.

It was three sets in union with one another.

You probably wanna pause this moment and just check that your set notation matches mine exactly.

Onto the second half of the learning now, where we're going to look at communicating solutions using set notation.

Sometimes, we'll be asked to communicate a solution set using set notation.

For example, solve six lots of x minus 3 is less than 54.

Give your answer using set notation.

We could solve that inequality in a wide variety of ways.

I'm gonna start by dividing both sides of the inequality by 6, adding +3 to both sides and there's my solution to that inequality, x is less than 12.

If we leave it here, we finish here.

We haven't completed our command.

We were told to give our answer using set notation so we wouldn't leave it written as an inequality, x is less than 12, we would have to write the set notation, x belongs to the set of values such that x is less than 12.

Quick check.

You've got that.

Laura tries solving this inequality.

Solve 5x minus 17 is greater than 23.

Give your answer using set notation.

She writes this solution.

What I'd like you to do is write a sentence of feedback to Laura telling her what she needs to improve.

Pause and write that sentence now.

You might have said, "You were asked to give your answer using set notation.

You left your solution written as an inequality.

Always read the question! You should have written, x belongs to set such that x is greater than 8." "Thank you," says Laura, "You are welcome." Solutions will include non-integer values and we can use those in set notation, too.

Let's have a look at an example, If we are asked to solve 1 is less than 3x minus 1 is less than x plus 12 and told to give our answer using set notation.

We'd start by solving 1 is less than 3x minus 1.

I'm gonna add +1 to both sides and I'll divide both sides by 3, x must be greater than 2/3.

Next we'd solve 3x minus 1 is less than x plus 12.

I will add -x to both sides, divide both sides by 2, x must be less than 13/2.

We can write the solution using set notation like so, x belongs to the set of values such that 2/3 is less than x is less than 13/2.

Your turn now to do similar example.

I'd like you to solve, <v ->10 is less than 5x plus 1 is less than x.

</v> I'd like you to give your answer using set notation.

Pause and try this problem now, don't be worried about getting it wrong.

I'll be back in a moment to check your work and we can correct any errors then.

Welcome back.

Let's see how we did.

Hopefully you started by solving <v ->10 is less than 5x plus 1.

</v> Let's add -1 to both sides and divide both sides by 5.

x must be greater than -11/5.

Next we're gonna solve 5x plus 1 is less than x.

Let's add -x to both sides and -1 to both sides and then we'll divide both sides by 4, x must be less than -1/4.

Are we finished? Absolutely not.

We've solved the inequality but we haven't given our answer using set notation.

You should have gone on to write your solution using set notation.

In set notation, we can also communicate information about the type of number the set contains.

Let's compare two examples.

There's two bits of set notation there, which values are in each set? I'd like you to pause and think about that for a moment.

Welcome back.

I wonder what you said, if you listed some values.

I wonder if you used words to describe each set.

I hopefully you noticed in the case of x belongs to the set such that -1 is less than x is less than 4.

There are infinitely many values in this set.

We couldn't possibly list them.

Yes, 0, 1, 2, 3 are in that set, but we've also got a half, a third, a quarter, a fifth, 0.

1, 0.

01, 0.

001.

I could go on listing infinitely the many values in that set, but in the case of this second example of set notation, hope you spotted that crucial difference because we know x is an integer.

There's only four values in the intersection of the sets.

0, 1, 2, and 3.

We can't have a half, we can't have a third.

We can't have 0.

1 because it wouldn't belong to the set.

x is an integer.

Quick check.

You've got that.

Which set notation correctly reflects these values.

Pause, have a look at the values represented on the number line and then decide which set notation is right to describe them.

Welcome back.

Hopefully, you said A.

Let's have a look at why it was not B or C.

In the case of B, that's not the right answer.

Because whilst we will see x is a positive integer used in set notation, in this case, it is incorrect because it would've excluded the value 0, -1, -2.

Our first set had to reflect the fact that x is an integer, not just a positive integer.

In the case of C, this set would've infinitely many values in it, not just the five integer values we were seeking.

Aisha wants to express these values using set notation.

This looks very similar to the example we just saw.

Aisha says, "I like this maths.

I can write x belongs to the set such that x is an integer in union with x belongs to the set such that -4 is less than or equal to x is less than or equal to 3." Laura says, "Almost Aisha, you have made one key error though." Can you spot Aisha's error? It's a tiny error but it makes a huge difference.

Pause, see if you can spot what's wrong here.

Welcome back.

Did you spot it? Using the union symbol is an error in this case.

If you use the union symbol, we've got an infinite set of values when we say x is an integer and we've got an infinite set of values when we say x is such that -4 is less than or equal to x is less than or equal to 3.

The union of these two infinite sets of values would be another infinite set of values, not the eight integer values we were seeking.

Let's think of it a different way.

Aisha was describing the two sets correctly.

x is an integer.

That would be all of the integer values on the number line.

x is such that -4 is less than or equal to x is less than or equal to 3 would look like so.

The values we want are the ones where these two sets of values intersect.

Hence, we use the intersection symbol to correctly write these values in set notation.

Quick check.

You've got this.

Which set notation correctly reflects these values? You've got some values on a number line there.

3 options to choose from.

Pause, see if you can pick out the right one.

Welcome back.

Hopefully you spotted was option B, so have a look at why it wasn't A or C.

In the case of A, that would be the union of two infinite sets of values that would make another infinite set.

That's not what we're looking for.

In the case of C, don't use words if you can say it more concisely with notation.

In the case of B, we are very concisely, accurately describing those values.

Practise time now.

For question one, we're solving inequalities, but read the questions carefully.

In each case we're asked to give our answer using set notation.

Be sure to do that having solved the inequalities.

Pause, try these 3 problems now.

Question two, which of the below represent this set of values? We've got six examples of set notation.

Some of them represent those values, some of them do not, pick the ones that do, and then where the notation does not represent those values, I'd like you to write a sentence for each of those examples to explain why that one does not represent this set of values.

Pause and try this now.

Feedback time.

We were solving inequalities and writing our answers using set notation for question one.

In the case of A, our solution was x is less than 20 as an inequality and in set notation it looks like so.

For B, we had two inequalities to solve.

We should have found that x is greater than 3 and x is less than 21/4.

Writing that answer using set notation would look like so.

Question one part c.

Hopefully you spotted this question was slightly different.

It includes the phrase x is an integer.

Be careful about that when we go to communicate our answer.

Solving was nice and simple, if you start by dividing both sides by 7 and -17, there's a solution as an inequality but it's not finished because you have to write it using set notation and reflect the fact that x is an integer.

You do that by writing those two sets.

Be sure to use the intersection symbol, not the union one.

But question two, I asked you to select which of the below represent this set of values.

You should not have selected A, B absolutely does, C does, D does not, E does and F does not.

I asked you to then go on and write a sentence to explain why the ones that do not represent the set of values do not represent them.

Let's have a look at that.

In the case of A, you might have written something along the lines of, this is an infinite set of values, not the four specific ones we want.

In the case of D, an infinite set of values in union with another infinite set of values, it was incorrect to use the union symbol there and for F, an infinite set of all the negative integers, in that case.

We're at the end of the lesson now, sadly, but we've learned an awful lot.

We've learned that we can represent a solution set using set notation.

In this example, the inequalities on that number line, rather than saying x is less than 3 or x is greater than 9, we can write it as the union of those two sets.

Hope you enjoyed this lesson as much as I have and I look forward to seeing you again soon for more mathematics.

Goodbye for now.