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A closed interval on a number line is a set of numbers where the endpoints are included in the interval. The notation for a closed interval uses square brackets, like this: \( [a, b] \). In a closed interval, both limits \(a\) and \(b\) are part of the interval, meaning if \(x\) is any number within this interval, then \(a \leq x \leq b\).

Why do we use closed intervals? Quite simply, they help us specify that the limits are "closed," or included as part of the set. This is opposed to open intervals, which use parentheses \( (a, b) \) and do not include the endpoints. For example, in the interval \([-\frac{6}{5}, -\frac{1}{2}]\), we include \(-\frac{6}{5}\) and \(-\frac{1}{2}\) within our range.

Endpoints are essential in defining any interval, especially closed intervals. They are the values at which the interval begins and ends on a number line.

To locate these endpoints:

• Convert fractions to decimal form if necessary for easier placement on the number line.
• Mark each endpoint clearly on the number line to define the start and end of the interval.

For the interval \([-\frac{6}{5}, -\frac{1}{2}]\), the endpoints are \(-\frac{6}{5}\), which equals \(-1.2\), and \(-\frac{1}{2}\), which equals \(-0.5\). These points delineate our boundaries on the number line.

Shading on a number line is a visual method to indicate which part of the number line is included in the interval. When you look at an interval like \([-\frac{6}{5}, -\frac{1}{2}]\), shading is done between and including the endpoints.

How to shade:

• Identify the section between the two endpoints.
• Draw a continuous line between the endpoints.
• Ensure the endpoints themselves are marked with closed circles to show inclusion.

This shaded part signifies all the numbers falling in between \(-1.2\) and \(-0.5\), including these endpoints themselves.

Representation on a number line gives a clear visual of an interval. It helps you understand the scope of numbers we are considering within a given interval.

Steps to represent \([-\frac{6}{5}, -\frac{1}{2}]\):

• Draw a straight horizontal line to signify the number line.
• Mark and label the endpoints \(-1.2\) and \(-0.5\) with closed circles to include them.
• Shade the section between these two endpoints.

By following these steps, you provide a visual depiction that communicates precisely which numbers are part of the interval.