### Video Transcript

Which of the following inequalities is represented by the diagram? Circle your answer. The options are 𝑥 is greater than or equal to negative three but less than four, 𝑥 is greater than negative three and less than four, 𝑥 is greater than or equal to negative three and less than or equal to four, or 𝑥 is greater than negative three and less than or equal to four.

First, we need to determine the endpoints of this line. And we can see that they are at negative three and four. This means that the line spans all of the 𝑥-values between negative three and four. Notice that negative three and four are the endpoints for all four options we were given.

We need to work out the correct inequality signs. To do so, we can look at the types of circle that have been used at the two endpoints. Notice that we have a closed circle at the lower endpoint, but an open circle at the upper end. A closed circle is used to represent a weak inequality, less than or equal to or greater than or equal to. It means that the value at this point is included in the interval, whereas an open circle is used to represent a strict inequality, strictly less than or strictly greater than. The value at this point is not included in the interval.

So as we have a closed circle at the lower end of the interval, this means that negative three is included. So we can write this part of the inequality as 𝑥 is greater than or equal to negative three. We can write this the other way round as negative three is less than or equal to 𝑥.

At the upper end of the interval, we have an open circle, meaning the value of four is excluded. So this can be represented by the inequality 𝑥 is less than four. These two separate inequalities can be combined into a single double-sided inequality. 𝑥 is greater than or equal to negative three but less than four.

Looking carefully at the four options we were given, we can see that the option which has the correct pair of inequality signs is the first one. So we found our answer. 𝑥 is greater than or equal to negative three but less than four. You could also read this inequality as negative three is less than or equal to 𝑥, which is less than four.