Problem 8 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Show the interval on a number line. $$ \left[-\frac{6}{5},-\frac{1}{2}\right] $$

Step-by-Step Solution

Answer

To represent the closed interval \(\left[-\frac{6}{5},-\frac{1}{2}\right]\) on a number line, first mark the positions of the endpoints \(-\frac{6}{5}\) and \(-\frac{1}{2}\) on the number line. Next, shade the region between the endpoints and use closed circles (filled-in circles) to show that the interval includes both endpoints. Finally, label the closed circles with their corresponding endpoint values, \(-\frac{6}{5}\) and \(-\frac{1}{2}\).

Step 1: Understand the given interval

The given interval is \(\left[-\frac{6}{5},-\frac{1}{2}\right]\), which is a closed interval because it uses square brackets. This means the interval includes both of its endpoints, \(-\frac{6}{5}\) and \(-\frac{1}{2}\).

Step 2: Identify the location of the endpoints on the number line

First, mark the location of both endpoints on the number line. To do that, divide the number line into equal parts. Place \(-\frac{6}{5}\) and \(-\frac{1}{2}\) at their respective positions on the number line. Since \(-\frac{6}{5}=-1.2\) and \(-\frac{1}{2}=-0.5\), place \(-\frac{6}{5}\) to the left of \(-\frac{1}{2}\).

Step 3: Represent the closed interval on the number line

Now that the positions of the endpoints are marked on the number line, represent the closed interval by shading the region between the endpoints. To show that the interval includes both of its endpoints, use closed circles (also known as filled-in circles) on the number line at \(-\frac{6}{5}\) and \(-\frac{1}{2}\).

Step 4: Label the endpoints

Finally, label both of the closed circles on the number line with the values of the endpoints. Write \(-\frac{6}{5}\) below the closed circle on the left, and write \(-\frac{1}{2}\) below the closed circle on the right. Now, the interval \(\left[-\frac{6}{5},-\frac{1}{2}\right]\) is properly displayed on the number line, with the endpoints marked and the region between them shaded.

Key Concepts

Closed Interval

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

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

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 Number Line

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.