Lesson video

Hello, Mr. Robson here.

Welcome to maths.

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

Well, we love solving linear inequalities and we love graphs.

So let's see what it looks like when we combine those two joys.

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

You'll understand what all of that means shortly.

Lots of keywords in today's lesson.

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

Region is a keyword that you might not be familiar with.

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.

There's two parts to our learning today.

We're gonna begin by graphing inequalities in one variable.

We know lots of ways to represent the equality x is greater than 4.

A very obvious way is algebraically.

That is algebraic notation for x is greater than 4.

We could use set notation, x belongs to the set such that x is greater than 4.

We could use a number line.

Number lines are a lovely visual representation.

There's another visual representation which is really mathematically useful.

Let's take a look.

We can also represent the solution graphically for x is greater than 4.

We start by considering where x is equal to 4.

We can define that with this line.

Every point on that line is where x is equal to 4 or the x coordinate of any point on that line is equal to 4.

We then need to consider in which region x is greater than 4.

Let's look at a point to the left of that line and a point to the right of that line.

In the point to the left of the line, x is equal to 2 and in the point to the right, x is equal to six.

So in trying to satisfy the inequality, X is greater than 4, well, that point does not.

Whereas that point does.

We can find many more coordinate pairs whereby x is greater than 4.

There's another one and another one and another one and another few and another lot and loads more.

There is also an infinite number of non-integer value coordinate pairs where x is greater than 4.

We can represent these infinite points whereby x is greater than 4 by shading the region.

There's that keyword, region.

What does it look like? It looks like that.

A region is an area that graphically represents the solution to an inequality.

We can label a region R and I'm sure you can understand why we use R.

Quick check you've got this.

I'd like you to fill in the blanks.

There's just two blanks.

What do you think is going in those spaces? Pause, have a think.

Welcome back, hopefully you said that we call this area a region and we can label it with the letter R like so.

There's an important difference between these two inequalities, x is less than 3, x is less than or equal to 3.

Algebraically, we use a difference symbol to differentiate between the two.

It's x is less than 3, x is less than or equal to 3.

Note the difference between the two symbols.

On a number line, we use a hollow dot versus a solid dot to represent the difference.

A hollow dot for x less than 3 and a solid dot for x is less than or equal to 3.

The hollow dot means x cannot be 3.

The solid dot means x can be 3.

We also see an important difference when we graph the two inequalities.

Can you spot the difference between the two? Pause, tell the person next to you or say it aloud to me on screen.

Welcome back.

I'm sure you spotted the difference.

A dashed line, a solid line.

This is important so you want to pause and make a note of this once I've explained it to you.

On the left hand side of our screen, x is less than 3.

The dashed line means x cannot be 3.

On the example, on the right hand side of the screen, X is less than or equal to 3 and a solid line means x can be 3.

Pause and make a note of those important points.

Quick check you've got that.

What I'd like you to do is match these four inequalities to these four graphs.

Pause and pair them up.

Welcome back, for A, you should have matched that one to x is less than 2.

For B, x is greater than or equal to 2.

For C, x is greater than 2, which means D must have been x is less than or equal to 2.

A dashed line for A meaning x cannot be 2.

A solid line in B meaning x can be 2.

For C, a dashed line x cannot be 2 and for D, a solid line x can be equal to 2 in that example.

We know that we can graph these two inequalities.

Y is less than 3 and y is greater than -4.

We know that we can write a solution to both inequalities as a single solution set.

y is less than 3 and y is greater than -4.

Well, we could write that as a single solution set.

<v ->4 is less than y is less than 3</v> and we're being a little more mathematically efficient.

We can graph this single set solution.

Let's have a look at how we graph that.

<v ->4 is less than y is less than 3.

</v> Well that's where y is equal to 3, a dashed line because we know we're going to represent when y is less than 3.

That line is where y is equal to -4.

A dashed line because we know we're going to want the moments when y is greater than -4.

And it must be this region.

The region which satisfies us inequality is now highlighted.

We can check that this is definitely the region that satisfies this inequality.

We check by picking any coordinate pair.

I've picked the coordinate pair 5, -2.

In this case I've got a y value of -2.

I'll check that versus my single solution set, <v ->2 is indeed greater than -4</v> and it is less than 3.

I know that I've got the right region.

In maths, we have loads of checking mechanisms. It's important that we use them.

Quick check you've got this, which inequality is represented here.

You've got four to choose from.

Pause and take your pick.

Welcome back, hopefully you said it's option C.

<v ->3 is less than or equal to y is less than 5.

</v> Why was it not A? Well the region is bound by the lines y equals 5 and y equals -3.

Not x equals 5 and x equals -3.

It wasn't B because our region has two limits that inequality only has one.

And it wasn't D because that was an incorrect interpretation of dashed and solid lines.

The symbols were the wrong way round.

Sometimes we have to draw a region which satisfies multiple inequalities.

I wonder what this is gonna look like.

We might be asked to shade the region satisfied by -3 is less than y is less than or equal to 4 and x is less than 5.

We know what this region looks like, that's the region.

We can now consider the second region x is less than 5.

Well that's where x is equal to 5.

So where is x less than 5? On the left hand side of that line and not on the right hand side.

We can check that that is indeed the right region by picking any coordinate pair.

In this case I've selected 3, -1.

x equals 3 and y equals -1, do those values satisfy a pair of inequalities? Absolutely they do.

I know I'm in the right region.

Quick check you've got that.

Which inequalities are satisfied by this region? Pause, take your pick.

Welcome back, I did say when I asked the question, inequalities plural.

So you should have picked more than one.

The inequalities which are satisfied by this region are A and B.

Practise time now.

For question one, there's four points marked on that graph.

What I'd like you to do for part A, tell me which of those points do not satisfy the equality x is less than 1.

For part B, for each of the coordinate pairs which does not satisfy the inequality, I'd like you to write a sentence explaining why they do not.

And finally for C, you'll shade the region which satisfies the inequality x is less than 1.

Pause and do this now.

Question two, I'd like you to write the inequality satisfied by these regions.

You just need to write one inequality for each of those four examples.

Pause and do this now.

Question three, these regions will satisfy multiple inequalities.

What I'd like you to do in each case is shade the region satisfied by those pairs of inequalities.

Pause and try this.

Finally, question four, we're going to get creative.

Part A given that x and y are integers, suggest inequalities that satisfy these six coordinate pairs and no more coordinate pairs.

For part B of this question, I've got a big challenge for you.

How many inequalities can you come up with to define this region? Good luck with that one.

Pause and do this now.

Feedback time now.

Question one part A, which these points do not satisfy the inequality? You should have selected those two points.

For part B, I ask you to write a sentence explaining why they do not satisfy the inequality.

You might have written, the less than symbol and dashed line means x can't be 1.

And for the coordinate pair 2, 0, x is greater than 1 and it can't be in order to satisfy this inequality.

For part C, shading the region which satisfies the inequality, x is less than 1.

You should have shaded that region there and it should be labelled R.

Question two, I asked you to write the inequality satisfied by these regions.

For A, you should have written x is less than 0.

For B, why is less than or equal to -2.

For C, x is greater than -1 and for D -3 is less than y is less than or equal to 3.

Be absolutely sure that you've differentiated correctly between the dashed lines and the solid lines and the correct symbol.

Question three, we were shading regions satisfied by multiple inequalities.

For A, your graph should look like so.

For B like so.

And for C you should have that on your graph.

Pause, just check that your solutions match mine.

Question four, some creativity here x and y are integers, we were suggesting inequalities that satisfy the six coordinate pairs and no more.

You might have said 0 is less than x is less than 4 and -3 is less than y is less than 0, but that wasn't the only answer.

There were four ways to describe the bounds in the x direction, there they are.

And there were four ways to describe the bounds in the y direction.

So how many possible combinations does that give us? There were 16 possible pairings that you could have picked from and 16 is the maximum number of solutions that we could come up with.

Onto the second half of our learning now.

We're gonna look at graphing inequalities in two variables.

I wonder how that's going to look different.

Let's take a look.

We can also graphing qualities that involve two variables.

Oh look, y equals 2x-3.

That's a y variable and an x variable.

That's the line y equals 2x-3.

All of the coordinate pairs on the line satisfy the equation, y equals 2x-3 when substituted.

What do I mean by that? For that coordinate pair -1, -5, when we substitute x equals -1 and y equals -5 into our equation, the equation is satisfied.

I can do that for the next coordinate pair or the integer value coordinate pair 0, -3 when I substitute in those values, the equation is satisfied and the same thing happens as we move upon that line.

Every point on that line has coordinate pairings which satisfy that equation.

What about these coordinate pairs? A little shift of those coordinate pairs.

What difference does it make? When I substitute in -1, -4, we do not satisfy the equation y equals 2x-3 nor do we when we substitute 0, - 2 or any of those other coordinate pairs on that graph.

None of the values satisfy the equation but there is something we notice.

Can you spot it? For every coordinate pair, the value of y is greater than that of 2x-3.

The y values on the left hand side, the value of 2x-3 is on the right hand side and in this case -4 is greater than -5.

And in the next case too, <v ->2 is greater than -3</v> and it happens in all cases.

0 is greater than -1, 2 is greater than 1, 4 is greater than 3.

There are infinitely many coordinate pairs whereby the value of y is greater than that of 2x-3.

Including all of the non-integer value pairs not shown here.

For the sake of efficiency we shade the region and call it y is greater than 2x-3.

Once we've shaded a region, we label it R and we can label this as the region y is greater than 2x-3 because that's true for every point in that region.

Izzy is asked to graphically represent y is less than 2x-3.

What I'd like you to do for this little check is to use substitution to show why these two coordinate pairs belong in the region y is less than 2x-3.

Pause, get substituting and show why those two coordinate pairs belong.

Welcome back, the coordinate pair 4, 2, that's x equals 4, y equals 2.

When we substitute those inequality, the inequality is satisfied.

For the next pair, 2, -4 substitute in an x value of 2, a y value of -4 and again the inequality is satisfied.

It's not just by substitution that you can see these two coordinate pairs belong in the region.

By comparison you can see that their values are less than when y equals 2x-3.

If y equaled 2x-3, the point would be on the line, but in the case of these coordinate pairs, the y coordinate value is too low.

Hence we can see y is less than 2x-3.

Another quick check.

Izzy then plots all of these coordinate pairs.

Whilst they all satisfy the inequality, what should she have done differently to be more efficient and accurate? Pause, answer that with the person next to you or say your answer aloud to me on the screen.

Welcome back, hopefully you said something along the lines of, shading the region is more efficient; it automatically includes all of the infinite non-integer value coordinate pairs.

Izzy only had the integer value pairs.

We want all the pairs that belong in this region.

Do notice we've labelled it R for the region and we've given the region a name y is less than 2x-3.

The same dashed and solid lines are used to denote the difference between less than or equal to and less than.

You see the contrast in these two examples, y is less than 2x-3 has a dashed line, y is less than or equal to 2x-3 has a solid line.

Meaning in that example y can be equal to 2x-3.

Let's check you can apply that knowledge.

What I'd like you to do here is match the inequalities to the graphs.

Four inequalities, four graphs, pause and get pairing.

Welcome back, hopefully you paired graph A with the inequality y is greater than 3 minus a half x.

Graph B you paired that 1 with y is less than or equal to 3 minus a half x.

For C y is greater than or equal to 3 minus a half x.

And for D, y is less than 3 minus a half x.

Izzy is solving this problem.

Shade the region that does not satisfy the inequality y is less than or equal to 5x -2.

And Izzy says "0, -2 and 1, 3 are coordinate pairs on the line." And plots them.

"The less than or equal to symbol, therefore a solid line." And draws a solid line.

And then concludes "The inequality y less than or equal to 5x-2, therefore this region." Can you spot Izzy's mistake.

It's ever so minor but it's mathematically very important.

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

Welcome back and well done if you spotted this one.

In mathematics, it's really important to read the question carefully.

Izzy had done a lot of excellent maths but look at the question does not satisfy.

Izzy shaded the region that does satisfy the inequality y is less than or equal to 5x-2.

She should have shaded the region that does not.

So instead what should she have drawn, a dashed line with the region on that side of the graph.

A dashed line because we can't include any of the infinitely many values whereby y is equal to 5x-2.

Quick check you've got this.

Which region does not satisfy the inequality y is less than or equal to 0.

3x-2.

6.

Three to choose from.

Pause, see if you can select the right one.

Welcome back, hopefully you said option A.

For B, this region did satisfy the inequality y is less than 0.

3x - 2.

6.

And for C, the region does not satisfy the inequality y is less than 0.

3x-2.

6.

Ever such a slight difference between option C and our correct answer option A and it's the dashed line versus the solid line.

If we're not satisfying the inequality, y is less than or equal to 0.

3x-2.

6, we cannot include the values on the line y is equal to 0.

3x-2.

6.

Hence the correct answer was A because it had the dashed line.

Practise time now.

Question one part A, show by substitution that these two coordinate pairs satisfy the inequality y is less than 3-2x.

And B rather than by substituting, I'd like you to show visually that the two coordinate pairs satisfy the inequality.

Finally, for C, you can shade the region that satisfies that inequality.

Pause and do all of those things now Question two part A.

I'd like you to shade the region that satisfies the inequality y is greater than or equal to x-4.

For part B, I'd like to shade the region that satisfies the inequality y is less than 1 minus a quarter x.

Pause and do this now.

Question three, I'd like to shade the region not satisfied by the inequality y is less than or equal to 3x-1.

Pause and try this now.

Feedback time, question one part A show by substituting that the two coordinate pairs satisfy the inequality.

so let's start with the coordinate pair -1, 1, substitute those values into our inequality.

One is less than 5 an absolute truth.

The inequality is satisfied.

Is it also the case for the coordinate pair 2, -4? Absolutely -4 is less than -1, inequality is satisfied.

We know that this is the right region.

We also know it's the right region visually.

In part B, I asked you to show visually that the two coordinate pairs satisfy this inequality.

Well if y we're equal to 3-2x, the points will be on the line but they're not.

The points are below the line, below the line in the y direction.

Meaning y is indeed less than 3-2x.

By comparing those points to the line y equals 2x-3, we can see that these y values are less than when y is exactly 2x-3.

Finally, for part C we're shading the region that satisfied the inequality, we should have shaded that region there with a dashed line.

Question two part A, we can draw the line y equals x-4 by using a couple of coordinate pairs.

And then to shade the region y is greater than or equal to x-4 would look like so.

For part B, a handful of coordinate pairs to draw line y is equal to 1 minus a quarter x but it's a dashed line because we don't want to be equal to 1 minus a quarter x, we need to be less than.

So that region there.

For question three shading the region not satisfied by the inequality y is less than or equal to 3x-1.

Those points are on the line 3x-1 and a dashed line and that region a dashed line because we can't include any values on the line, y is equal to 3x-1.

And we shaded that region because we want the values not satisfied by the inequality.

Sadly, we at the end of the lesson now, but we've done all sorts of wonderful learning.

What have we learned? We've learned that we can represent linear inequalities graphically.

We know their solutions are a region which we shade to reflect the infinite number of coordinate pairs within.

A dashed line is used when graphing inequalities involving less than or greater than.

And a solid line for inequalities involving less than or equal to and greater than or equal to.

We can justify by substitution or visually why a point satisfies a given inequality.

And we know to be careful around the wording of whether a region satisfies or does not satisfy an inequality.

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

Goodbye for now.