Problem 50 - Precalculus

Question

Express each interval using inequality notation and show the given interval on a number line. $$[0, \infty)$$

Step-by-Step Solution

Answer

The interval \([0, \infty)\) in inequality notation is \(x \geq 0\).

Step 1: Understand the Interval

The interval \([0, \infty)\) includes all real numbers starting from 0 and extending to infinity. Note that 0 is included in the interval (indicated by the bracket \([\)) and infinity is not included (indicated by the parenthesis \()\)).

Step 2: Convert to Inequality Notation

For the interval \([0, \infty)\), we want to express this range of numbers using inequalities. In this interval, the variable \(x\) is greater than or equal to 0. The inequality notation will be: \(x \geq 0\).

Step 3: Illustrate on a Number Line

Draw a number line. Place a solid dot at 0 to indicate that 0 is included in the interval. Then, draw a line extending to the right towards infinity. Since infinity is a concept rather than a number, the line simply continues indefinitely to the right without an end point.

Key Concepts

Interval Notation

Interval notation is a compact way of representing a range of numbers, often used in mathematics to indicate intervals on the real number line. It uses brackets and parentheses to show whether endpoints are included or excluded.
Bracket symbols:

• \([\) - means the endpoint is included (closed interval).
• \(()\) - means the endpoint is not included (open interval).

For example, in the interval \([0, \infty)\), \(0\) is included, denoted by \([\), while infinity \((\infty)\) is never included, indicated by \()\).
Understanding this notation is crucial because it succinctly conveys which numbers belong to the interval, often simplifying the expression of constraints and solutions in mathematical problems.

Number Line Representation

A number line is a visual tool used to represent real numbers and their relationships to one another. When representing an interval on a number line, use visual clues to show which numbers are included and which are not.
For the interval \([0, \infty)\), place a solid dot at 0 to show it's included. From that point, draw an arrow or a line extending to the right, representing all numbers greater than 0. This line doesn't stop, reflecting the boundlessness towards infinity.
Number lines are useful because they provide a clear, intuitive overview of an interval, helping to easily visualize the solution to an inequality.

Real Numbers

Real numbers encompass all the numbers we usually work with, including integers, fractions, and irrational numbers. They form a continuous, unbroken set of values on the number line. When expressing intervals, we are often referring to a set of real numbers that satisfy certain conditions.
The interval \([0, \infty)\) includes every real number from 0 onward. This interval excludes negative numbers since they do not satisfy the condition \(x \geq 0\).
Understanding real numbers and their properties enables us to navigate problems involving continuous data and solve equations that model real-world phenomena.

Infinity Concept

Infinity is a concept used to describe something without limits, often in terms of size, quantity, or length. In mathematics, infinity is not a number but an idea used to explore limits and unbounded quantities.
In interval notation, \(\infty\) signals that there is no upper bound. For instance, in \([0, \infty)\), the interval extends indefinitely beyond any finite number.
This concept is essential because it allows mathematicians to describe infinitely large sets and understand behaviors approaching extremes, such as in calculus or topology. Embracing the abstract nature of infinity helps in appreciating the boundless potential of mathematical exploration.