Problem 7 - College Algebra with Modeling and Visualization

Question

Express each of the following in interval notation. $$ \\{x |-1 \leq x\\} $$

Step-by-Step Solution

Answer

\([-1, \infty)\)

Step 1: Analyze the Inequality

The given inequality is \(-1 \leq x\), which means that \(x\) can be any number greater than or equal to \(-1\). The inequality is inclusive of \(-1\) because of the 'less than or equal to' symbol (\(\leq\)).

Step 2: Understand Interval Notation

Interval notation is a way of representing a range of values that \(x\) can take. If the interval is inclusive of a value, we use a square bracket \([\) or \(]\). If it is exclusive, we use a parenthesis \(()\) or \())\). Since \(-1\) is included, we'll use \([\).

Step 3: Construct the Interval

Since \(x\) can be any number greater than or equal to \(-1\) and there is no upper bound given (\(x\) can extend to infinity), the interval notation for this inequality will be \([-1, \infty)\). The use of \(\infty\) always involves a parenthesis because infinity isn't a specific number that can be included in the interval.

Key Concepts

Introduction to Inequalities

Inequalities are mathematical expressions that describe the relationship between two values or expressions using inequality symbols. These symbols include:

• Greater than (>),
• Less than (<),
• Greater than or equal to (≥),
• Less than or equal to (≤)

In the exercise provided, we deal with an inequality \(-1 \leq x\), which means any number greater than or equal to \(-1\). Understanding inequalities is crucial as they tell us which numbers satisfy the given constraints. Solving such expressions often involves finding a range of values for the variable that makes the inequality valid, offering insight into how variables can behave within certain limits.

Exploring Infinity

Infinity is a concept that extends beyond any number you can count. It's not an actual number but more like an idea representing something without an end. In mathematics, it is denoted by the symbol \( \infty \). When used in interval notation, infinity indicates that there is no upper or lower boundary to the values a variable can take.

In the exercise, the notation \([-1, \infty)\) means starting from \(-1\) (inclusive) and extending indefinitely towards larger numbers. It's important to know that whenever infinity is mentioned, the associated interval notation uses a parenthesis () not a bracket [], because infinity isn't a defined, reachable point.

Understanding Algebraic Expressions

Algebraic expressions are mathematical phrases that can contain numbers, variables, and operation symbols. They are used to represent real-world problems in simpler terms.

In our exercise, the algebraic expression \(-1 \leq x\) is an inequality representing values of \(x\) that are either equal to or greater than \(-1\). This type of expression is essential for forming equations and inequalities used in algebra.
They help us describe and solve problems relating to quantities and their relationships. When you learn to manipulate these expressions, you build a foundation for solving more complex mathematical problems.