Problem 34 - Algebra and Trigonometry

Question

\(33-38\) . Express the interval in terms of inequalities, and then graph the interval. $$ (2,8] $$

Step-by-Step Solution

Answer

Interval: \(2 < x \leq 8\); Graph with open circle at 2, closed at 8.

Step 1: Identify Interval

The given interval is \((2,8]\). This interval includes all numbers greater than 2 and less than or equal to 8.

Step 2: Write Inequality

The interval \((2,8]\) corresponds to the inequality \(2 < x \leq 8\), where \(x\) is the variable representing the numbers within this range.

Step 3: Graph Interval

To graph the interval \((2,8]\), draw a number line. Use an open circle at 2 to represent that 2 is not included and a closed circle at 8 to show that 8 is included. Shade the region between these two points to indicate all numbers in this interval.

Key Concepts

Inequalities

Inequalities are mathematical expressions used to show the relationship between two values when they are not equal. They are represented using symbols like \(<\), \(>\), \(\leq\), and \(\geq\). These symbols indicate how values compare to each other:

• \(<\) means "less than".
• \(>\) means "greater than".
• \(\leq\) means "less than or equal to".
• \(\geq\) means "greater than or equal to".

When we examine the interval \((2,8]\), we convert it into an inequality to show the range of numbers it represents. In this case, the inequality is expressed as \(2 < x \leq 8\). This means \(x\) can be any real number that is more than 2 but less than or equal to 8.

Graphing Intervals

Graphing intervals involves depicting the set of numbers that satisfy an inequality on a number line. This visual representation helps easily identify which numbers are included in the interval and which are not.
For the interval \((2,8]\), follow these steps:

• Draw a horizontal line, this is your number line.
• Place an open circle at 2 to show that 2 is not included in the interval. Open circles signify that the endpoint is not part of the solution.
• Place a closed circle at 8 to denote that 8 is included in the interval. Closed circles mean the endpoint is included.
• Shade the region between the circles to highlight that all the numbers in between are part of the interval.

This method helps anyone understand at a glance which values satisfy the inequality \(2 < x \leq 8\).

Open and Closed Intervals

Intervals can be classified into open or closed based on whether their endpoints are included in the set of numbers. This classification helps understand how to use circles when graphing and how to write them in inequality form.
In open intervals, depicted with parentheses such as \((a,b)\), the endpoints \(a\) and \(b\) are not included in the interval. This is why open circles are used for graphing.
Closed intervals, shown with brackets like \([a,b]\), mean both endpoints are included in the interval. Closed circles are used in graphs to represent this inclusion.
The interval \((2,8]\) combines both open and closed properties: it does not include 2 (making it open at 2), but includes 8 (making it closed at 8). Thus, it uses both an open circle at 2 and a closed circle at 8 on the number line.