Lesson video

Hello, my name is Ms. Parnham.

And in this lesson, we're going to learn how to solve quadratic inequalities, where A is greater than one.

When we say solving quadratic inequalities, where is greater than one, we mean that the X squared coefficient is greater than one.

So in this example, we're going to solve eight X squared plus two X, subtract three is greater than zero.

And the first thing we're going to do is factorised it.

Now you can use whatever factorising method you prefer for quadratics, where you have an X squared coefficient greater than one.

In this example, we'll use what's known as the AC method.

So we're looking for a sum of two and we've taken two from the X coefficient and the product of negative 24.

That's the product of eight and negative three.

The combination that gives us this is negative four and positive six.

So that's how we're going to split up two X.

So if we just factorise the first two terms, and then factorise the second two terms, this gives us four X multiplied by two X subtract one, plus three multiplied by two X subtract one greater than zero.

And when we put that into two brackets, we have four X plus three multiplied by two X subtract one greater than zero.

Now we're going to rough out a quick sketch graph of this curve.

It is a parabola because it is a quadratic.

It's based on X squared.

And if the first bracket was zero, four X would have to be negative three, and therefore, we've got a root of negative three quarters.

And if the second bracket was zero, two X would have to be one.

So we have a root of a half there.

So these are the two crossing points.

All off the X axis.

And we just put a rough parabola on there.

Now this is greater than zero, so we are talking about above the X axis.

So this is two distinct sections of the graph.

So before we get to negative three quarters, and after we pass through the half.

So X is less than negative three quarters or X is greater than a half.

And we can show this on a number line.

If we just place a circle on 0.

5, which is equivalent to a half, X is greater than this.

So the arrow points to everything greater.

And then a circle over a negative 0.

75, the decimal equivalent of negative three quarters, and the arrow points to everything smaller than that.

The circles are left unshaded because the inequalities are greater than and less than respectively.

I'm using set notation because these two sections are distinct.

We need the union symbol in between the separate inequalities.

Here's a question for you to try.

Pause the video to complete the task, and restart the video when you're finished.

Here are the answers.

Use whatever method you prefer to factorise the quadratic.

And you will also need to use your estimation skills when placing the circle for negative two-thirds above the number line because these increments are marked off in one fifth or 0.

2 intervals.

Here are some more questions for you to try.

Pause the video to complete the task, and restart the video when you're finished.

Here are the answers.

So the quadratic expression on the left of the inequality symbol, when you factorise that into brackets, for part A you should get two X plus one multiplied by two X plus five.

That can be in either order.

Remember, multiplication is commutative.

For part B you should have got four X subtract three, multiplied by two X plus three.

And then for part C, five X subtract two, multiplied by five X add two.

Perhaps you notice that there is the difference of two squares that you are using to factorise that expression.

Here's a further question for you to try.

Pause the video to complete the task, and restart the video when you're finished.

Here are the answers.

This question supported you to manipulate the inequality until we have zero on one side of the less than symbol.

And then we just solve as before.

Here are some more questions for you to try.

Pause the video to complete the task, and restart the video when you're finished.

Here are the answers.

In question five, you're working backwards.

So, from the set notation you know that X is greater than four, or it's less than one.

So that means we're talking about two distinct sets.

Therefore, we have X subtract four in one bracket, X subtract one in the other bracket.

And that's greater than 0 because that's above the X axis, because of it being in those two distinct sets.

That's all for this lesson.

Thank you for watching.