Problem 5 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Show the interval on a number line. $$ (3,6) $$

Step-by-Step Solution

Answer

First, draw a number line and mark the boundary points 3 and 6. Use open circles to indicate that these points are not included in the interval. Then, draw a line connecting the open circles to represent the interval (3,6). Finally, label the interval with the notation "\( (3, 6) \)" above the line.

Step 1: Draw a number line

Draw a horizontal line and label it as the number line. Make sure to include enough space to represent the interval of interest (3,6).

Step 2: Mark the boundary points

Place two marks on the number line, one representing the point at 3 and the other at 6. Label these points as '3' and '6' respectively.

Step 3: Indicate the boundary points are not included

Since the interval (3,6) does not include the boundary points, we'll represent them using open circles. Draw an open circle above the point labeled '3' and another above the point labeled '6'.

Step 4: Draw the interval

Draw a straight line connecting the open circles above the points '3' and '6'. This line represents the interval of all numbers between 3 and 6.

Step 5: Label the interval

Write the interval notation, "(3,6)", above the line connecting the open circles. Now, you have successfully represented the interval (3,6) on a number line.

Key Concepts

Number Line Representation

Understanding the principles of number line representation is essential in visualizing numerical intervals. A number line is a visual representation of numbers along a straight line where each point corresponds to a real number. To illustrate an interval like \((3,6)\), we would begin by drawing a horizontal line—the number line itself.

To accurately represent the interval \((3,6)\) on the number line, we cherry-pick two specific points: 3 and 6. These selected points are crucial because they delineate the range of our interval. Consider these points as your starting and ending markers; everything that falls between them is part of the interval we're interested in. When it comes to delineate these points, it's important to know whether they are included in the interval, which leads us into our discussion about boundary points and open intervals.

Boundary Points

Boundary points on a number line are the critical markers that define the limits of an interval. In interval notation, boundary points are represented by the numbers that mark the start and the end of the interval. However, how we draw these points on the number line depends on whether the interval is inclusive or exclusive of these points.

In our example interval \((3,6)\), the numbers 3 and 6 are boundary points. And here's where the nuanced details matter—since the interval notation uses parentheses \(()\), it signifies that the interval is open, meaning the values at 3 and 6 are not included in the interval. On a number line, this is indicated by drawing open circles at these points rather than closed dots, which would suggest inclusion. Hence, grasping the significance of boundary points and the symbols that represent them is key in conveying precise mathematical meanings.

Open Intervals

An open interval is a range of numbers between two boundary points, excluding the boundary points themselves. In our example \((3,6)\), the interval is 'open' because 3 and 6 are not part of the solution set. Open intervals are critical in mathematics as they represent a continuum of values an expression can take, except for precisely defined edges.

When we represent open intervals like \((3,6)\) on a number line, we ensure not to include the boundary points. As per the exercise solution steps, we draw open circles at the points corresponding to 3 and 6 and connect them with a line, excluding the boundary points, to capture all possible values that fall strictly in-between. This visual aid greatly assists students in understanding how open intervals function and emphasizes the importance of carefully reading mathematical notation for proper interpretation.