Problem 42 - Algebra and Trigonometry

Question

\(41-46=\) Express the interval in terms of inequalities, and then graph the interval. $$ (2,8] $$

Step-by-Step Solution

Answer

The inequality is \(2 < x \leq 8\) and the graph is a line between 2 (open) and 8 (closed).

Step 1: Understand the Interval Notation

The interval \((2, 8]\) uses parentheses and brackets to define which endpoints are included or excluded. Specifically, in this interval, 2 is excluded (open parenthesis), and 8 is included (closed bracket).

Step 2: Convert Interval to Inequality

Write the interval \((2, 8]\) as an inequality. Since 2 is not included, we use the inequality \(x > 2\). Since 8 is included, we use \(x \leq 8\). So, the inequality is:\[2 < x \leq 8\]

Step 3: Set Up Graphing Mechanism

When graphing, we use a number line. Mark the positions for 2 and 8 on this number line.

Step 4: Graph the Interval

Draw an open circle at 2 to represent that 2 is not included in the interval, and draw a closed circle at 8 to represent that 8 is included. Shade the region between them, indicating that all numbers between but not including 2, and up to 8 including 8, are part of the interval.

Key Concepts

Understanding Inequalities

Inequalities are mathematical expressions that show the relationship between two values when they are not equal. Instead of using an equal sign, inequalities use the symbols \(<\), \(>\), \(\leq\), and \(\geq\). These symbols help us indicate if one value is smaller or larger than the other.
For an interval like \((2, 8]\), we translate this to an inequality. The inequality tells us that the variable lies between two values: 2 and 8 in this case. The inequality derived from this interval is \(2 < x \leq 8\), where:

• \(2 < x\) means any value greater than 2.
• \(x \leq 8\) means any value less than or equal to 8.

Notice the difference between the symbols \(>\) and \(\leq\). The symbol \(>\) means the number is not equal to 2, while \(\leq\) indicates that 8 is included.

Open and Closed Intervals

An interval can be classified as open, closed, or a combination of both. This classification is determined by whether the endpoint is included in the interval or not.

• **Open Interval**: An open interval does not include its endpoints. It is represented by parentheses \(()\). For example, in \((2, 8)\), both 2 and 8 are excluded.
• **Closed Interval**: A closed interval includes its endpoints. It is denoted by brackets \([])\). For instance, \([2, 8]\) includes both 2 and 8.
• **Half-Open/Half-Closed Interval**: This is a mix of open and closed. They include one endpoint and exclude the other, like \((2, 8]\), which excludes 2 but includes 8.

In our case of \( (2, 8]\), you should picture a space between 2 and 8 where 2 is not counted in, but 8 is.

Graphing on a Number Line

Graphing an interval on a number line is a visual way to represent inequalities.
First, draw a horizontal line and mark points representing numbers on this line, concentrating on the values that define our interval.
For the interval \((2, 8]\):

• Place an open circle at 2 to show that 2 is not included.
• Place a closed circle at 8 to indicate that 8 is included.

Then, shade the region between these two points. This shaded area indicates all the numbers that satisfy the inequality \(2 < x \leq 8\). Thus, graphing helps visually understand which numbers are part of the interval and which ones aren't.