Write (22,98] in set-builder notation.
The Correct answer and Explanation is:
The interval (22,98](22, 98](22,98] can be expressed in set-builder notation as:{x∣22<x≤98}\{ x \mid 22 < x \leq 98 \}{x∣22<x≤98}
Explanation
Set-builder notation is a concise way to describe a set by specifying the properties that its members must satisfy. The notation consists of a set symbol, usually braces {}\{ \}{}, followed by a variable (commonly xxx) and a vertical bar (or colon) that means “such that.” After the bar, we include a description of the properties or conditions that elements in the set must fulfill.
In this case, the interval (22,98](22, 98](22,98] consists of all numbers greater than 22 and up to and including 98. Let’s break down the interval:
- Open and Closed Ends:
- The parenthesis “(” next to 22 indicates that 22 is not included in the set. Therefore, any value that is equal to or less than 22 does not satisfy the conditions of the set.
- The bracket “]” next to 98 indicates that 98 is included in the set, meaning any value up to and including 98 is part of this interval.
- The Set Definition:
- The set is defined as containing all values xxx that meet the condition 22<x22 < x22<x. This means xxx can take any value larger than 22 but must also satisfy x≤98x \leq 98x≤98. Thus, the second part of the condition ensures that xxx does not exceed 98.
- Examples of Elements in the Set:
- Values such as 22.1, 50, 97.9, and 98 are in the set because they meet the criteria. However, values like 22, 22.0, and 99 are not included in the set since they do not satisfy the specified conditions.
In summary, using set-builder notation effectively captures the essential characteristics of the interval (22,98](22, 98](22,98], allowing for a precise mathematical description that is clear and unambiguous.