Problem 49 - Precalculus

Question

Express each interval using inequality notation and show the given interval on a number line. $$[-1, \infty)$$

Step-by-Step Solution

Answer

The inequality is \(x \geq -1\).

Step 1: Understand the Interval

The interval \([-1, \infty)\) represents all real numbers starting from \(-1\) and extending to infinity. The interval includes \(-1\) but does not include \(\infty\).

Step 2: Write the Inequality

Based on the interval \([-1, \infty)\), the inequality can be expressed as \(x \geq -1\), meaning \(x\) is greater than or equal to \(-1\).

Step 3: Draw the Number Line

On a number line, mark the point \(-1\) with a closed dot (indicating inclusion of \(-1\)) and shade the line extending to the right towards positive infinity. This visualization reflects the set of all numbers greater than or equal to \(-1\).

Key Concepts

Number Line

A number line is a visual representation of numbers on a straight line. It helps us understand and compare numbers by showing their positions relative to each other. Each point on the number line corresponds to a real number.

Features of a number line include:

• A point labeled as zero, which serves as the center or origin of the line.
• Numbers increasing positively to the right and negatively to the left.

For interval \([-1, \infty)\), we use closed and open dots to demonstrate inclusion and exclusion. A closed dot at -1 means -1 is included, while an open dot would indicate a number is not included. However, for infinity, the number line cannot be infinitely long, so we use an arrow pointing to the right to symbolize that the interval goes on indefinitely without a finite endpoint.

Real Numbers

Real numbers (\( \mathbb{R} \)) consist of all rational and irrational numbers. They include various subsets like natural numbers, whole numbers, integers, and decimal numbers. When considering intervals like \([-1, \infty)\), all elements included are real numbers.

• Rational Numbers: These can be expressed as the quotient of two integers, such as \(\frac{3}{4}\).
• Irrational Numbers: These cannot be neatly expressed as a simple fraction. Examples include \(\sqrt{2}\) or \(\pi\).

Real numbers provide a comprehensive backdrop for intervals because they encompass almost every numerical value one could think of, except for imaginary numbers.

Interval Notation

Interval notation is a convenient way to describe sets of numbers, especially for real number intervals. It uses brackets to show the start and end of an interval.

Here's how it works:

• A square bracket \([\ or ]\) means that the boundary number is included in the interval.
• A parenthesis \((\ or )\) signifies that the boundary is not included.

For example, in the interval \([-1, \infty)\), the square bracket \([\) indicates that -1 is included in the interval, while the parenthesis \()\) shows infinity is not included. This is because infinity is a concept rather than a specific number that can be reached or measured.