Lesson video

Hello, Mr. Robson here.

Welcome to Maths.

Super choice to join me today, especially because we're solving a set of linear inequalities graphically.

This is a pretty awesome bit of mathematics, so let's not hang around here.

Let's get stuck in.

Our learning outcome is we'll be able to extend our thinking to multiple linear inequalities with two variables, and show that this can be solved graphically to find a solution set.

Some keywords we'll hear today, inequality.

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

And the word region.

A region is an area that graphically represents the solutions to one or more inequalities.

Every coordinate pair within the region satisfies the inequalities that define the region.

Two parts to our learning.

Let's begin by solving multiple inequalities graphically.

We can graph regions that satisfy inequalities such as y is greater than x and y is less than four.

To graph the region where y is greater than x, we can start by graphing the line y equals x.

And then looking at a coordinate pair, in this example, one, four.

If x equals one and y equals four, we found the region which satisfies that inequality.

We'll highlight that region.

This is the region where y is greater than x.

What about y is less than four? We'll start with the line y equals four and we'll highlight the region where the y coordinate values are less than four.

We'll label it R, this is our region, y is less than four.

We could also graph one region that satisfies both inequalities simultaneously.

Let's have a look at what that means.

Shade the region that satisfies the inequalities y is greater than x and y is less than four.

We'll start by graphing the lines again.

There's y equals x, there's y equals four.

The two lines divided our grid into four possible regions.

I could label them region one, region two, region three, and region four.

We can test the regions to see which one satisfies both inequalities.

We can test the region R1.

Need a coordinate pair from that region so we can pick the coordinate pair six, five.

I'm gonna substitute those x and y values into our inequalities.

When we substitute them into the first inequality, we find the inequality is not satisfied.

The same thing happens for the inequality y is less than four.

Five is not less than than four.

The inequality is not satisfied so the region R1 satisfies neither inequality.

Let's test R2.

Any coordinate pair within the region is useful for the test.

I've gone for the coordinate pair one, five.

When we substitute it into y is greater than x, we find that inequality is satisfied, however, the second inequality is not.

R2 satisfies only one of our inequalities.

Let's test R4.

Any coordinate pair from in that region will do.

I've gone for three, zero.

When I substitute that into y is greater than x, I find that inequality is not satisfied.

However, the second inequality is, but it's not good enough.

R4 satisfies only one of our inequalities.

When we test R3, we notice something importantly different.

Any coordinate pair from within R3, one, three, I've gone four.

When I substitute that into our two inequalities, I find y is greater than x is satisfied and y is less than four is satisfied.

R3 is the region which satisfies both inequalities simultaneously.

When we're asked to shade the region that satisfies both inequalities simultaneously, we would just shade that region.

We're just shading the one region whereby all coordinate pairs contained within which satisfy both inequalities simultaneously, so your graph will look like so.

Quick check you've got this.

The lines y equals negative two and y equals x plus three divide our grid into four regions.

What I'd like you to do is test a point from each region to show which region satisfies both of these inequalities simultaneously.

Pause, do those tests now.

Welcome back.

You're welcome to test them in any order.

I'm gonna test region four first with a coordinate pair zero, negative five.

Substitute them into our two inequalities and I find that our first inequality is not satisfied.

Our second inequality is satisfied, but R4 satisfies only one of our two inequalities.

Testing R3, I use a coordinate pair of negative seven, negative three.

When I substitute those values in, I find the first inequality is not satisfied and the second inequality is not satisfied also.

So we can say that R3 satisfies neither inequality.

Testing R2, I use the coordinate pair zero, five.

When I substitute that into our first inequality, it's satisfied but the second inequality is not.

So at the region R2, we can say it satisfies only one inequality.

R1, I'm using the coordinate pair two, zero.

When I substitute that in, the first inequality and the second inequality are satisfied, so we can say R1 satisfies both inequalities simultaneously.

We might have to satisfy three inequalities simultaneously.

For example, shade the region that satisfies the inequalities y is less than x plus three, y is less than or equal to four minus x, and y is greater than zero.

We'll start with lines.

There y is equal to x plus three, y is equal to four minus x, and y is equal to zero.

You'll notice where I've used dashed lines because it was a less than sign in the inequality or a soft line when it's allowed to be equal to.

We can see the region that satisfies all three of these inequalities.

What do I mean by that? We can see the region where y is less than the line x plus three.

We can see the region where y is less than or equal to four minus x, and we can see where y is greater than zero.

We then shade the region where all three of those inequalities were satisfied.

It is this region which satisfies all three inequalities simultaneously.

But it's always good to check.

Pick any coordinate pair from within the region.

In this example I've gone for the coordinate pair one, one.

We substitute into our first inequality, second inequality, and third inequality and we find that all three are satisfied.

We often get checking mechanisms in mathematics and in this lesson it's no different.

We absolutely want to use them to check that we're correct.

Quick check you've got this.

I'd like you to shade the region that satisfies the inequalities y is less than two x, y is less than five minus x, and y is greater than or equal to negative two.

Remember what I said about checking mechanisms? Be sure to check your answer.

Pause, give this problem a go now.

Welcome back.

The line y equals two x, dashed to reflect inequality.

The line y equals five minus x, dashed to reflect that inequality, and the solid line y equals negative two because y is allowed to be equal to negative two.

And we shade that region.

How do we know it's that region? We test.

You could have used any coordinate pair to test.

I'm gonna use one, zero, they're nice simple numbers to substitute in.

When I substitute in the first inequality, second inequality, and third inequality, I see that they're all satisfied.

This must be our region.

The region satisfies all three inequalities simultaneously, I can conclude.

Using any coordinate pair in this region to check is absolutely fine.

You didn't have to use the pair one, zero.

Jun is solving this problem.

Shade the region that satisfies the inequalities y is greater than x, y is greater than negative x, and y is greater than three.

Jun says, these are definitely the correct lines, so it must be the region that forms a triangle in the middle.

Jun has made at least one error.

Can you see why? Pause, see if you can spot what's gone wrong in this example.

Welcome back.

A few things that we could point out to Jun.

We might have said, Jun, you've got the wrong region.

We could also have said, Jun did not check his region.

If we checked for the coordinate pair, we would've known that we haven't got the correct region.

We use the example zero, one.

We'll find it satisfies the first inequality, satisfies the second inequality, but does not satisfy the third.

By checking, we know there's an error.

Jun was right about something though.

These are definitely the correct lines, y equals x, y equals negative x, y equals three.

Dashed lines for all three to reflect those inequalities.

So what's gone wrong? We might want to try and help Jun visualise the region in this case.

That would give us the region where y is greater than x.

There, y is greater than negative x and it's here crucially that y is greater than three.

It was in fact this region up here that satisfied all three inequalities simultaneously.

Of course this is the region.

Thank you, says Jun.

You're welcome, we'd say.

Quick check you've got this.

Shade the region that satisfies the inequalities y is less than x, y is greater than negative x, and y is less than three.

Pause, see if you can spot that region now.

Welcome back.

Hopefully you shaded that region over there.

How do we know? We check any coordinate pair from within that region.

Two, zero seem like nice simple values to substitute in to me.

Substitute them into our first inequality, it's satisfied.

In our second inequality, satisfied.

Also those values satisfy our third inequality.

We know we've got the right region.

The region satisfies all three inequalities simultaneously.

The important thing to note here is that just because the lines formed a triangle doesn't necessarily make that triangle the region we are seeking.

We have to think a little deeper than just going for the obvious answer in this case.

Practise time now.

Question one, we're gonna shade the regions that satisfy these pairs of inequalities.

Two questions there for you to have a go at.

Pause and try them now.

Question one, part C and D.

Similar problem but slightly different inequalities.

Part C and part D look incredibly similar.

You'll have to pay careful attention to how they are different.

Pause and do that now.

Question two, we're gonna shade regions that satisfy three inequalities for both parts A and B.

Again, they look like incredibly similar questions but there are subtle important differences so pay careful attention to the detail.

Pause, try these two now.

For question three, part A, I'd like you to show why this region does not satisfy these three inequalities.

Then for part B, you're gonna correct this error.

You're gonna shade the correct region that does satisfy those three inequalities.

Pause and do this now.

Feedback time, let's see how we did.

Question one, part A, shade in the region that satisfies those two inequalities.

We should have drawn those two dashed lines and shaded that region.

For part B, we should have those two dashed lines and that region.

Just pause and check that your work matches mine.

Part C and D, very similar questions, subtly and importantly different.

Part C should look like so, those dashed lines with that region highlighted.

Part D is the same two dashed lines, but importantly a different region highlighted.

Again, pause and check that your work matches mine.

Question two, we were shading regions that satisfy three inequalities simultaneously.

You should have those three lines, two which are solid, one which is dashed, and that region in the middle there for part A.

Part B looked like an incredibly similar question and it was an incredibly similar question.

It involved the same three lines on our graph but the region was different.

Again, pause and check that your work matches mine.

For question three, part A, we were showing why this region does not satisfy the inequalities.

So you can pick any coordinate pair in there.

I've gone for zero, negative one on account that they're nice, easy numbers to substitute.

When I substitute them into the first inequality it's satisfied.

The second inequality is not satisfied, the third inequality is satisfied.

So we've shown that this is not our region.

Shading the correct region would've looked like so.

Onto the second half of our learning now where we're gonna be solving graphically using technology.

Modern technology allows mathematicians to solve complex problems quickly, easily, and accurately provided the technology is correctly programmed.

So let's have a look at what it means to correctly programme technology for this topic in mathematics.

We'll start by going to desmos.

com.

Once there, we're going to click on Graphing Calculator.

After clicking upon Graphing Calculator, your screen should look like this.

Your next click is gonna be the Graphing Settings icon.

That's that one in the top-right there.

You'll name click the right one because your screen will look like this.

From here I'd like you to deselect minor gridlines.

Turn those off.

And I'd like you to type in a step of one on both the y axis and the x axis.

Next I'm gonna ask you to practise zooming.

Using these icons you can zoom in and out.

There's also ways to zoom on your mouse and using a touch screen if you have one.

I'd like you to practise that skill now.

We're also gonna need to look at various regions within the graph, so I'd like you to practise scrolling.

Again, you can do scrolling with your mouse or with your fingers on a touch screen.

Practise that skill now.

We're now ready to ask Desmos to graph inequalities.

This is wonderfully simple.

I'd like you to type y is greater than two x minus three here.

Pause and do that.

Welcome back.

Wasn't that wonderful? Desmos has graphed the inequality for us.

Desmos quickly and easily shows us the region which satisfies this inequality.

Quick check you can do an example just like that.

I'd like you to use Desmos to graph the region which satisfies the inequality y is less than two minus three x.

Pause and do that.

Welcome back.

Your screen should look like so.

You'll notice typing y is less than two minus three x in here graphed you that region.

Desmos can also graph the inequality y is less than or equal to two minus three x.

Didn't we just do that one? No, we graphed y is less than two minus three x.

Crucially, this is y is less than or equal to two minus three x.

So we need to change our less than symbol to a less than or equal to symbol.

A few ways we can do that.

We can click the keyboard icon.

Clicking the keyboard icon on the bottom-left there brings up our keyboard options, one of which is the less than or equal to symbol.

So you could retype that inequality introducing that symbol.

When you do that, you notice we get the correct region.

It's now a solid line and not a dashed line to represent the fact that values can be on that line.

Alternative method would've been to type the less than symbol immediately followed by the equals button.

If you type them one after the other, Desmos will recognise this as less than or equal to and automatically input that symbol for you.

So pause this and try typing that inequality using that method too.

Quick check you've got this.

I'd like you to use Desmos to graph the region which satisfies the inequality y is less than or equal to one minus x.

Pause and do this.

Welcome back.

Hopefully your screen looks just like that and you've learned that by typing y is less than or equal to one minus x in here will graph that region.

It must be a solid line marking the boundary of the region to correctly reflect that inequality.

Desmos can graph a region which satisfies a set of linear inequalities simultaneously.

For example, shade the region that satisfies the inequalities y is greater than x and y is less than four.

We can type y is greater than x on one line and y is less than four on the next.

But I think you spot the same problem I spotted when we do this.

The problem is that Desmos automatically shades the entirety of both regions.

The reader would need to note that it's the slightly darker shaded region which is the one that we want.

They may easily be mistaken into thinking that anything shaded is in our region, whereas we know that's not the truth.

There are two ways that Desmos enables us to get around this problem.

It's the same problem, we're shading the region which satisfies the inequalities y is greater than x and y is less than four.

And we'll begin by typing the region y is greater than x.

We can add y is less than four as a domain restriction on the same line using braces.

That will look like so.

Have a go at typing that in now.

Welcome back.

Hopefully your screen looks like mine and it looks almost finished.

There's a little bit more detail we need to add to make this look correct.

We have the region but we don't have the boundaries correctly defined.

There should be a dashed line on y equals four there.

We can draw the lines y equals x and y equals four to overcome that.

Crucially, if we click and hold here, we can turn solid lines into dashed ones to make sure our inequality is properly represented.

Quick check you've got that.

Sam is trying to graph this region.

The region that satisfies inequalities y is less than two x and y is greater than negative four.

Sam has made an error.

What advice would you give them to enable the tech to draw the region correctly? Pause and see if you can come up with some advice for Sam.

Welcome back.

You might have spotted the error was in that line there on Desmos.

And you might have said to Sam, you need to write the second inequality in braces.

If we do that and your Desmos entries look like that, the region is correctly drawn.

It's not just by using braces that we can overcome this over-shading problem.

Let me show you a second way.

We can graph each line as a function.

Rather than graphing those dashed lines as y equals x and y equals four, we can graph them as f of x equals x and g of x equals four.

We now want the region where y is greater than the f of x function and less than the g of x function.

We can get that by typing in that inequality, f of x is less than y is less than g of x.

Pause and see if you can make your screen look just like mine.

Welcome back.

Hopefully like me, using functions, we've been able to graph this one region that satisfies both inequalities simultaneously.

Quick check you've got this.

Which of the following would successfully graph the region that satisfies inequalities y is greater than negative x and y is less than x? Three options to choose from.

Which one is gonna work? Pause and do this now.

Welcome back.

Hopefully you spotted its option A.

Why is it not B? Well in the case of B, the f and g are the wrong way round.

We want the region where y is greater than the g of x function and less than the f of x function, not as it's laid out in B.

For C, that would've shaded solutions to both inequalities, not just the one region where both are satisfied simultaneously.

If you correctly type that into your desmos.

com, you'll find your screen looks like so.

It's wonderfully efficient this method, but it has its limitations.

We can only graph two inequalities using this method and also we can't graph vertical lines.

That might hold us back on some examples that we're looking to explore.

If we want to graph three inequalities, we need to use braces.

For example, we can shade the region that satisfies all three of these inequalities simultaneously.

We'll begin by graphing the lines, in this case they look like so.

And then we need to type in the region.

You'll notice my first inequality, I've just typed.

My second and third inequality, I've put inside braces.

If you type that into Desmos, your screen will look like so.

We've graphed a region that satisfies all three inequalities simultaneously.

Quick check that you can do that.

I'd like you to shade the region that satisfies these three inequalities simultaneously.

Pause and give it a go now.

Welcome back, let's see how we did.

Hopefully your screen looks just like this and you should have typed that into Desmos as well as the three lines dashed of course.

You needed your regions represented like so.

One of the inequalities typed normally, two of them in pairs of braces.

Practise time now.

Question one, I'd like you to graph the region which satisfies all three of these inequalities simultaneously.

For part B of that question, I'd like to know what the area of the shaded region is.

Pause and do this problem now.

Question two, we're gonna graph the region which satisfies these three inequalities.

For part B of this question, if x and y are integers, I'd like you to list the coordinate pairs that lie within the region.

Pause and try this problem now.

Feedback time.

Let's see how we did.

Graphing the region using Desmos should have looked like so.

We should have typed that to get our region one inequality followed by two inequalities in pairs of braces.

And in terms of the area of the shaded region, it's a triangle with an area of 10 by five over two.

That's 25 square units.

Question two, our Desmos screen should look like so.

Do note we've got two dashed lines, one solid line as the boundary of this region.

You should have typed that to get that region.

For part B, if x and y are integers listing the coordinate pairs that line within the region.

We could have any one of these coordinate pairs.

That would've been this list and your list should have contained 12 coordinate pairs.

We're at the end of the lesson now, sadly, but we've learned that we can graph multiple linear inequalities with two variables to show a solution set.

We know after finding a region to test any coordinate pair within it to show that our region satisfies each inequality.

We know that just because the lines form a triangle or any other shape, it's not necessarily the area inside the shape which solves all inequalities simultaneously.

Hope you've enjoyed this lesson as much as I have.

I look forward to seeing you again soon for more Mathematics.

Goodbye for now.