Lesson video

Hello, my name is Miss Parnham, and in this lesson, we're going to learn how to solve simple quadratic inequalities.

Let's start with a simple example of X squared less than or equal to 36.

We will square root both sides of the inequalities in order to isolate X, but if we square roots 36, we have two solutions to this, and we can see that from our sketch graph.

So obviously, root of 36 is six, but also we have a solution of negative six.

So we have X is less than or equal to six.

You can see the inequality is exactly the same as was in the simple quadratic inequality, but because of the nature of X squared where we have symmetry, we have X, which is greater than or equal to negative six.

We can actually show this on a number line.

So the solution is somewhere between six and negative six inclusive, and you can see we've shaded in the circles on this number line diagram to show that it can be equal to six or negative six, as well as in between them.

Here are some questions for you to try, pause the video, to complete the task, and restart the video when you're finished.

Here are the answers, when we square root 16, this has of course two solutions of four and negative four.

And if we have X less than four, then it must be X is greater than negative four.

So opposite signs and opposite inequality symbols.

The circles on the diagram must be left unshaded to show that X can be equal to four or equal to negative four.

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.

When we square root a number, notice that the inequality with the positive solution has the same inequality as X squared, and the inequality with the negative solution has exactly the opposite inequality symbol.

This time we'll look at an inequality that needs some manipulation first.

So we have three X squared add four greater than 79.

So if we're going to isolate X, let's subtract four from both sides.

Three X squared is greater than 75 and divided by three gives us X squared is greater than 25, and now we can square root.

Now we remember from before that square root in a number, gives us two solutions, a positive one and a negative one, and we use the symmetry of the X squared graph.

So we know that X is greater than five, so the positive solution has the same inequality, but the negative solution X is less than negative five, the inequality is reflected.

We're going to represent this using set notation, but before we do that, let's see it as a diagram on a number line.

This will help us understand why this is different from the previous example.

So we have a circle over five because X is greater than five, and we're pointing to everything that's greater than that.

It's unshaded because it doesn't include five, and a circle over negative five, and X is less than negative five, so the arrow points to everything less than that.

Again, negative five is not included.

This is not all one complete line, it's two separate sections.

So when we use set notation, we need something to join up these two inequalities, and we use the union symbol in between.

So we have X is greater than five, union X is less than negative five.

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.

Where we have a continuous set of solutions, the set notation contains a double inequality with X in between the negative and positive bounds.

Here is 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.

Where we have two separate sets of solutions, so the set notation uses the union symbol to show the two possible inequalities combined.

That's all for this lesson, thank you for watching.