Lesson video

Hello, Mr. Robson here.

Well done for making the right decision to join me for maths today.

It's gonna be a super lesson because we're solving quadratic inequalities graphically.

Well, let me check.

I love quadratics.

I love inequalities.

I love graphs.

I love solving things.

So this should be a lot of fun.

Let's take a look.

Our learning outcome is I'll be able to solve a quadratic inequality graphically.

Some key words we'll hear today: an inequality is used to show that one expression may not be equal to another.

A quadratic is an equation, graph or sequence where the highest exponent of the variable is 2.

The general form for a quadratic is a x squared + bx + c.

There's two parts to our learning today and we're gonna begin by solving graphically.

We can solve quadratic equations such as this one: x squared - 4 = 0.

There's lots of ways we might solve this.

One of which is to rearrange it to x squared = 4, and then take square root of both sides.

In this case, we wanna take both the positive and negative root of 4.

We get x = -2 and x = 2 as our solutions.

This quadratic equation has two solutions.

But what does the solution to a quadratic inequality look like? x squared - 4 is less than 0, what do you think the solution's gonna look like? Would it be the same as x squared - 4 = 0? Or will it be different somehow? Pause and have a think about that problem.

I'll be back in a moment to tell you more about it.

Welcome back, I hope you conjectured that they're going to be different.

The solution to x squared - 4 is less than 0 has to be different to the solution of x squared - 4 = 0, but how will it be different? Let's take a look.

What does a solution to a quadratic inequality look like? There's a reason why I've highlighted the word look.

Look is the keyword here because we can see the solution to this inequality when we graph it.

That's the curve y = x squared - 4.

What this inequality is asking us is when is the graph of x squared - 4 below 0? Can you see the solution now? It's here: x squared - 4 is below 0 here.

We can define this as a region.

It's when x is greater than -2 and less than 2.

We can graph that region x is greater than -2 and less than 2.

In fact, that region is our solution.

The solution to x squared - 4 is less than 0 is -2 is less than x is less than 2.

Quick check, you can do an example like that one.

Use the graph to solve x squared - 1 is less than 0.

Pause, try this problem now.

Welcome back, hopefully you said option b.

In the region x is greater than -1 but less than 1, the graph of x squared - 1 is less than 0.

We can solve a variety of quadratic inequalities using the same graph.

In this example, we're gonna use that one curve to solve all three of these inequalities.

Starting with x squared - 4 is less than 0, that's asking us where is the curve below the x-axis.

Well, that will be there In this area, what do we call this area? We call it the region x is greater than -2 but less than 2.

We can graph that region to show the solution to x squared - 4 is less than 0.

Next, where is the curve below 5? It's below 5 here, in this region.

We can define that region as x is greater than -3 and less than 3.

We can graph that region like so.

This last one is a problem.

Can you see why? x squared - 4 is less than -4.

There is the line y = -4.

The minimum point of our curve is -4.

That means that inequality has no solution because our curve will never be below -4.

We can't solve every quadratic inequality.

Quick check, you've got this.

I'd like you to solve x squared + 2 is less than 6.

I have provided a graph to help.

Pause, find the solution to this problem now.

Welcome back, I hope you said c.

When x is greater than -2 and less than 2, that's when the curve is below 6.

Another problem for you to try.

I'd like you to solve x squared + 2 is less than or equal to 1.

Pause, have a think about this problem.

Welcome back, did you spot that this problem is indeed a problem? It's option d.

There's no solution.

There's the line y = 1.

The minimum point of our curve is 2.

The curve will never be equal to or below 1.

Hence, there's no solution to this inequality.

We will see very complex-looking examples.

For example, solve -x squared + x + 6 is greater than 4.

It's quite an intimidating-looking problem.

But when we have the graph, this one is no more complex than the examples we've already done.

There's the curve of -x squared + x + 6.

When is the curve above 4? It's in that region and we can define that region as x is greater than -1 and less than 2.

There, it wasn't so bad after all.

Quick check, you can do a similar problem.

I provided the graph for you.

Using that graph, you should be able to solve -x squared + x + 6 is greater than -6.

Pause and have a go at this one.

Welcome back, I hope you rephrased that inequality to when is the curve greater than -6? Well, that happens here, in this region when x is greater than -3 and less than 4, well done.

Sometimes, the solution is slightly more complex.

Solve x squared - 4 is greater than 0.

Haven't we seen this one already? It looks awfully familiar, or is it different? It's very different.

This inequality is asking us when is the curve greater than 0? Well, the curve is greater than 0, the point where I've put those arrows on the graph.

It's in these two regions.

So the solution is x is less than -2 or x is greater than 2.

That's when the curve x squared - 4 is greater than 0.

We might see a solution like that, written using set notation rather than the word OR.

To rewrite that solution in set notation, it would look like so: x belongs to the set x is less than -2 in union with the set x is greater than 2.

Quick check, you've got that.

I'd like you to use this graph to solve x squared - 1 is greater than 0.

Four options to choose from.

Pause, see if you can figure it out.

Welcome back, hopefully, you said options c and d.

The curve is greater than 0 when x is less than -1 or it's greater than 1.

You would, of course, look to communicate that answer using set notation as well as inequalities.

Hence, option c and d are the solutions.

We can also solve a double inequality: x squared - 4 is greater than 0 and less than 5.

That's asking us when is our curve greater than 0 but less than 5? Well, that's going to be in these two areas or these two regions.

When x is greater than -3 but less than -2 and x is greater than 2 but less than 3.

Again, you might see such a solution written using set notation.

Quick check, you can do a similar problem to that one.

I'd like you to use this graph to solve x squared - 4 is greater than -3 and less than 0.

Pause, have a think about this problem.

Welcome back, hopefully, you turned that inequality into the question when is the curve greater than -3 but less than 0? And used this visual reference on the graph.

It's in these two areas here or should I say those two regions? The regions x is greater than -2 and less than -1 or x is greater than 1 and less than 2.

Did you rewrite that solution in set notation? If so, it should look like that and well done to you.

Practise time now.

For the question one, I'd like you to use the graph to solve these inequalities.

Five inequalities and you can solve them all using this graph.

Pause and do that now.

Question two, I'd like you to use this graph to solve these inequalities.

Four inequalities all solvable with just this one graph.

Pause, give this one a go.

Feedback time now.

Let's see how you got on with question one.

For part a, solution to that inequality was x is greater than -3 or less than 3.

For part b, x is greater than -4 and less than 4.

For part c, there was no solution.

When will that curve be below -9? Never, -9 is the minimum point of the curve, so that one has no solution.

For part d, x is less than -3 or x is greater than 3.

Hopefully you had to go at writing that solution using set notation like so.

Similar solution to part e but a little more complex again, x is greater than -4 but less than -2 or it's greater than 2 and less than -4.

Again, writing that solution using set notation should have looked like that.

Let's see how we did with question two.

For part a, x is greater than 2 and less than 5.

For part b, x is greater than 0 and less than 7.

For part c, x was less than -2 or x was greater than 5.

You would've written that solution using set notation also.

And for d, x is greater than 1 or less than 3 or x is greater than 4 and less than 6.

That solution using set notation looks like so.

Onto the second half of our learning now, we'll be solving by considering the roots.

Considering the roots of a curve can be useful when solving quadratic inequalities.

We saw in part one of the lesson, the solutions to these two inequalities.

We know that this curve x squared - 4 has roots at - 2 and +2.

The curve is below 0 between these roots.

It's a positive parameter, so it's below 0 between the two roots.

Hence, the solution to x squared - 4 is less than 0 is x is greater than -2 and less than 2.

The curve is above 0 outside of these roots.

Hence, the solution to our second inequality is x is less than -2 or x is greater than 2.

It's useful to check these solutions.

How do we check them? By substitution, if we're saying that the solution to the first inequality is x is greater than -2 and less than 2, then x = 0 should satisfy the inequality.

So we plug x = 0 into our inequality: 0 squared - 4, is that less than 0? It is indeed, inequality satisfied.

For our second inequality, if we're saying x is less than -2 or x is greater than 2, when we substitute in x = -4 and we substitute in x = +4, the inequality should be satisfied and it is.

It's really useful in algebra to check your work.

Quick check, now you can do a similar problem to that one.

This is the graph of x squared - 144.

By considering the roots, solve these two inequalities.

Pause and do this now.

Welcome back, you should have identified that the curve has roots at -12 and +12.

In the first case, we're asking when is the curve below 0? Well, it's below 0 when x is greater than -12 or less than 12.

When is the curve above 0? Well, that's when x is less than -12 or x is greater than 12.

You would, of course, check your answers.

If our solution to the first inequality is correct, then x = 0 should satisfy that inequality.

0 squared - 144, is that less than 0? It is indeed.

We've checked that solution.

We can check the second solution: x = -13 and x = +13 should satisfy our inequality and those values do, always useful to check your work.

We may not always be given a graph but solving graphically may still be our best option.

For example, if we were asked to solve x squared - 7 is less than 2, there's lots of ways we could approach this problem.

Rearranging it to x squared - 9 is less than 0 and sketching a graph is highly efficient.

We just added -2 to both sides of the inequality to rearrange it to x squared - 9 is less than 0.

What's useful about that rearrangement? Well done, we can sketch it and we know that there's roots at -3 and +3.

Our region must be x is greater than -3 and less than 3.

That's when the curve x squared - 9 is below 0.

That was a really efficient way to solve x squared - 7 is less than 2, always useful to check our solution, of course.

If that solution is correct, then x = 0 should satisfy the inequality.

I'm gonna substitute x = 0 into the original inequality x squared - 7 is less than 2.

So 0 squared - 7, is that less than 2? It is indeed.

We can validate our solution by checking.

Let's see if you can do a similar problem now.

Rearrange the form x squared + b is less than 0 and graph to solve.

A sketch will do when you graph this one.

Pause, try this problem now.

Welcome back, your rearrangement should have looked like so.

Add -4 to both sides of the inequality, x squared - 16 is less than 0.

A sketch with roots at -4 and +4, when is our curve below 0 in that region there when x is greater than -4 and less than 4? Of course, we want to check our solution.

If our solution's correct, then x = 0 should satisfy our original inequality 0 squared - 12 is less than 4.

Is that satisfied? It is indeed.

We've checked our solution.

We have multiple methods, for example, like this one.

Solve 7 - x squared is less than -2.

We could rearrange it into 9 - x squared is less than 0.

By adding +2 to both sides of the inequality, I get this rearrangement.

The sketch of which will look like so.

That's a sketch of 9 - x squared with roots at -3 and +3.

Now when is the curve below 0? It's gonna be below 0 when x is less than -3 or x is greater than 3.

We might write that solution set in set notation.

We can check if that's correct by substituting in -4 and +4 and we know that we've satisfied the original inequality we've checked our answer.

I said we have multiple methods.

We do, let's have a look at a different rearrangement.

7 - x squared is less than -2.

If we add +x squared and -7 to both sides of that inequality, we get this rearrangement: 0 is less than x squared - 9.

The curve, our sketch of a very different appearance this time, but it's still got roots -3 and +3.

This time we're asking, when is that curve above 0? Funnily enough, it's the exact same regions.

We've got the exact same solution, multiple methods giving us the exact same solutions for that one inequality.

Quick check, you can do a similar problem.

Which rearrangements could we use to solve this inequality? 37 - x squared is less than 12.

Four to choose from, which ones are gonna work? Welcome back, I wonder what you said? Did you go for option a? Well done, adding -12 to both sides of the inequality gives us 25 - x squared is less than 0.

Graphically, it looks something like that.

Roots at -5 and +5, when is that curve below 0? When x is less than -5 or x is greater than 5, that solution in set notation also.

Did you spot the second rearrangement? Well done, option d.

By adding +x squared and -37 to both sides of the inequality, we get to option d.

The graph looks very different.

We've got a positive parabola this time, but the same roots, -5 and +5.

When is that curve above 0? Funny enough, it's the exact same solution.

Another quick check: Aisha is solving this inequality.

40 - x squared is less than 4.

Aisha says: I can rearrange this to 36 - x squared is less than 0.

The curve has roots at 6 and -6.

The solution is x is greater than -6 and less than 6.

Aisha has made an error.

Give her some feedback as to what she could improve.

Pause, say that feedback to the person next to you or maybe write it down.

Welcome back, I wonder what you said? Hopefully, you said something along the lines of check your solution.

The solution that Aisha has given us is x is greater than -6 or less than 6, so x = 0 should work.

But when we substitute that into our original inequality, 40 - 0 squared is less than 4, no, it's not.

Our solution is not correct.

You might also have said sketch the graph to visualise the solution.

When you get to the point 36 - x squared, a sketch like so is really powerful, we can see roots at -6 and +6.

When is the curve below 0? Well, that would be when x is less than -6 or x is greater than 6.

Sketches, visual representations are hugely powerful in mathematics especially so in this topic.

Practise time now.

Question one, by considering the roots, solve these two inequalities.

Pause and do this now.

Question two, part a, I'd like you to rearrange this equality to the form x squared + b is greater than 0.

I'd like you to sketch the curve of your rearrangement and then solve that inequality.

For part b, I'm gonna practise our fluency.

You're going to show me two different rearrangements, two different sketches, to solve that one inequality.

Essentially, you're doing part b twice.

The reassuring thing is you are going to get the same solution both times.

You'll know you are right.

Pause, try these problems now.

Welcome back, feedback time.

Question one, we were considering the roots to solve these two inequalities.

They were roots at -9 and +9.

When is the curve below 0? Well, that's when x is greater than -9 and less than 9.

We can check that by substituting in x = 0 and it does indeed satisfy our inequality.

For part b, when is the curve above 0? It's in those two regions, when x is less than -9 or x is greater than 9.

You might have written your solution and set notation.

I do hope you checked it.

Any value less than -9, I'm using x = -10.

Any value greater than 9, I'm using x = +10.

They both satisfy the inequality.

We know those solutions are correct.

Question two, part a, I asked you to rearrange this inequality.

You should have added -19 to both sides to get the arrangement x squared - 64 is greater than 0.

The sketch would've looked like so with roots at -8 and +8.

When is our curve above 0? It's above 0 when x is less than -8 or x is greater than 8.

Should have written that solution using set notation also.

Question two, part b, I asked you to do two different rearrangements, two different sketches, to solve this inequality.

Method one would be to add -17 to both sides of the inequality to get 100 - x squared is greater than 0.

Your sketch would've looked like so with roots at -10 and +10.

A second rearrangement would've been to add +x squared and -117 to both sides of the inequality to get x squared - 100 is less than 0.

Your sketch would've looked like so.

When is that curve less than 0? Well, that's between those two roots.

Both methods, both sketches reveal the solution x is greater than -10 and less than 10.

We're at the end of the lesson now, sadly.

We have learned that we can solve quadratic inequalities graphically.

We know that there is a difference between x squared - 9 is less than 0 and x squared - 9 is greater than 0.

We can check our solutions by substitution to confirm their validity.

We can also use sketches of the graphs to support our understanding of the solution.

It's a lovely piece of math today, wasn't it? Hope you enjoyed it as much as I did.

I'm going to look forward to seeing you again very soon for more mathematics.

Goodbye for now.