Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Write an inequality that represents the interval. Then state whether the interval is bounded or unbounded. \((2,10]\)

Step-by-Step Solution

Answer

The inequality representing the interval \((2,10]\) is \(2 < x \leq 10\). The interval is bounded.

Step 1: Conversion of Interval to Inequality

In the interval \((2,10]\), the number 2 is not included and the number 10 is included. Hence, the corresponding inequality will be \(2 < x \leq 10\).

Step 2: Determine if the interval is bounded or unbounded

The given interval \((2,10]\) has a lower limit and an upper limit; these are 2 and 10 respectively. These limits bound the numbers that \(x\) can represent in the inequality. Therefore, the interval is bounded.

Key Concepts

Interval Notation

Interval notation is a method used to describe sets of numbers that fall within a certain range. It is commonly used in mathematics to specify the solution sets for inequalities. The interval notation for the set of all real numbers, x, that are greater than 2 but less than or equal to 10 is written as
(2,10]. The parentheses, '(', indicate that the number 2 is not included in the set, while the square bracket, ']', signifies that 10 is included. Similarly, if both endpoints were included, square brackets would be used on both ends [a,b]; if neither endpoint were included, the interval would be denoted with parentheses: (a,b).
Understanding how to read interval notation allows you to quickly grasp the range of possible values in a mathematical context. When an inequality is expressed in interval notation, it provides a clear visual representation of the smallest and largest possible values within the range.

Bounded Intervals

Bounded intervals are intervals that have both a lower limit and an upper limit. In the example
(2,10], the interval is bounded because it has a clear start at number 2 and ends at number 10. Bounded intervals can be of two types: closed or open. A closed interval is one where both the starting and ending numbers are included in the range, denoted by square brackets [a,b]. An open interval, denoted by parentheses (a,b), does not include the endpoints.

Closed and Open Bounded Intervals

Understanding the distinction between open and closed is essential when dealing with bounded intervals. For instance, [2,10] is a closed interval including all numbers from 2 to 10, while (2,10) is an open interval that includes all numbers greater than 2 and less than 10, but not 2 or 10 themselves. The interval
(2,10] is a semi-open interval since it contains all numbers greater than 2 up to and including 10.

Inequality Notation

Inequality notation is used to represent a range of values that satisfy a particular condition, usually with greater than or less than signs. It's a way to mathematically express the relationships between numbers. For example, the interval
(2,10] can be written in inequality notation as

\(2 < x \leq 10\).

This notation reads as 'x is a number greater than 2 and less than or equal to 10'. The inequality sign '<' indicates that the number 2 is not included in the solution set, while the sign '\leq' (less than or equal to) indicates that 10 is included.

Translating Between Notations

Understanding how to convert between interval notation and inequality notation is a valuable skill. It allows students to represent solutions in multiple ways, facilitating a better understanding of the concept. Whenever faced with interval notation, remember that parentheses correspond to strict inequalities (>, <) and brackets correspond to inclusive inequalities (\geq, \leq). Ensuring you remember this can help avoid mistakes and allow you to communicate mathematical ideas effectively.