Problem 66 - Algebra for College Students

Question

Give, in interval notation, the unknown numbers in each description. A number is between -3 and -2 .

Step-by-Step Solution

Answer

The interval is \((-3, -2)\).

Step 1: Understand the Interval Notation

Interval notation is a way to describe the set of numbers between two endpoints. The notation uses brackets and parentheses to show whether the endpoints are included or excluded. For example, \((-3, -2)\) means all numbers between -3 and -2, but not including -3 and -2 themselves.

Step 2: Determine the Endpoints

In this problem, the endpoints are given as -3 and -2. Since the question states that the number is between -3 and -2, we will use parentheses, meaning that the endpoints themselves are not included.

Step 3: Write in Interval Notation

To express the numbers between -3 and -2 in interval notation, we use parentheses to exclude the endpoints. Therefore, the interval notation for this problem is: \((-3, -2)\).

Key Concepts

Algebraic Notation Basics

Algebraic notation is a way to represent mathematical concepts in symbols and variables. It's commonly used to express equations, inequalities, and relationships between numbers. For example, an algebraic expression like \(x + y = 5\) means that the sum of \(x\) and \(y\) is equal to 5. When dealing with intervals, algebraic notation helps to describe a range of numbers. If we say \(-3 < x < -2\), it translates to any number that is greater than -3 and less than -2.

Understanding Endpoint Inclusion

Endpoints define the boundaries of an interval. Whether they are included or excluded affects the interval notation.
To **include** an endpoint, we use brackets, like in \([-3, -2]\). This means both -3 and -2 are part of the interval.
Let's say the interval is inclusive of its endpoints, then a number can be -3, -2, or any value in between. Conversely, excluding them uses parentheses, such as \(-3, -2\). Here, -3 and -2 are not included. Only values strictly between -3 and -2 are considered part of this interval.
Understanding endpoint inclusion is crucial for correctly interpreting and writing interval notations.

Concept of Interval Exclusion

Interval exclusion means that the endpoints are not considered part of the interval. We demonstrate this with parentheses instead of brackets.
For example, if we exclude the endpoints -3 and -2, our interval notation looks like \((-3, -2)\). This notation indicates that values between -3 and -2 are included, but -3 and -2 themselves are not.
When applying interval exclusion to solve problems, always check the context to ensure you correctly interpret which numbers are included or excluded. It's a key detail that shifts the meaning and solution significantly. By mastering interval exclusion, you enhance your ability to describe and understand numerical ranges properly.