Problem 13 - College Algebra

Question

In Exercises 13-26, express each interval in terms of an inequality and graph the interval on a number line. $$(1,6]$$

Step-by-Step Solution

Answer

The interval \( (1,6] \) is expressed as the inequality \( 1 < x \leq 6 \). The graph consists of a line connecting 1 and 6, with an open circle at 1 and a filled circle at 6.

Step 1: Identify the type of interval

The interval given is \( (1,6] \). Considering the given symbols, it signifies that the interval begins just after 1 (as 1 is not included) and extends up to and including 6.

Step 2: Convert to inequality

The inequality representation of \( (1,6] \) is written as: \( 1 < x \leq 6 \). This can be read as: 'x is greater than 1 and less than or equal to 6'.

Step 3: Graph the interval

On the number line, locate and mark the points 1 and 6. Draw a line connecting them. In accordance with the symbols in the interval, make an open circle at 1 (to show that 1 is not included) and a filled (or closed) circle at 6 (to show that 6 is included).

Key Concepts

Graphing Intervals

Graphing intervals visually represents a range of numbers on a number line. This helps to easily understand which values are included in a set, and which are not. A point on a number line corresponds to each number we are interested in.

• To start graphing, identify the endpoints of the interval. For the interval \((1, 6]\), these endpoints are 1 and 6.
• Draw a number line and mark the positions of these endpoints.
• Next, use circles to denote whether the endpoints are included in the interval. An open circle indicates that the number is not part of the interval, whereas a closed circle means that it is included.

For our interval \((1, 6]\), 1 has an open circle, while 6 has a closed one, visually demonstrating which endpoints are included or excluded.

Open and Closed Intervals

The concepts of open and closed intervals are key to understanding how intervals are defined in mathematics. These terms indicate whether the endpoints of an interval are part of the interval.

• Open Interval: An interval is considered open when it does not include its endpoints. For example, the interval \((1,6)\) includes all numbers between 1 and 6 but does not include 1 and 6 themselves. This is represented on a number line with open circles on both ends.
• Closed Interval: A closed interval includes its endpoints. For example, the interval \([1,6]\) includes both 1 and 6 as well as every number in between. On a number line, this is shown with closed circles at each end.
• Half-open (or half-closed) Interval: Sometimes, an interval may be open on one end and closed on the other. In our example \((1,6]\), it is open at 1 and closed at 6. This is a simple combination of the open and closed formats.

Understanding these types helps you determine inclusion or exclusion of endpoints within an interval when graphing.

Number Line Representation

Representing intervals on a number line is a straightforward way to visualize inequalities. This method allows you to "see" the range of values quickly.

• A number line is a straight line where numbers are placed according to their value. Negative values are on the left, while positive ones are on the right, with zero typically in the center if included.
• To display an inequality like \(1 < x \leq 6\) on a number line, you start by marking the relevant points — in this case, 1 and 6.
• The solution set of the inequality is shown by drawing a line or segment between these two points. This indicates the continuous range that solutions can fall into.
  • An open circle at 1 indicates that it is not included in the solution set. A closed circle at 6, however, shows that it is included.

Number line representation is especially useful when dealing with inequalities, as it provides a clear picture of the relationship between values and helps confirm which numbers satisfy the given constraints.