Problem 51 - Precalculus

Question

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

Step-by-Step Solution

Answer

The interval \((-\infty, 1)\) is expressed as \(x < 1\) in inequality notation and shown on a number line with an open circle at 1, shading to the left.

Step 1: Understand the Interval

The given interval is \((-\infty, 1)\). This interval includes all numbers that are less than 1. The round parenthesis means that the number 1 is not included in the interval.

Step 2: Express in Inequality Notation

To express the interval \((-\infty, 1)\) in inequality notation, you state that any number \(x\) within the interval satisfies the condition \(x < 1\).

Step 3: Draw the Number Line

To represent the interval on a number line, draw a horizontal line and locate the point 1 on it. Shade the line to the left of 1, indicating that all numbers less than 1 are included in the interval. Use an open circle at 1 to show that 1 is not included.

Key Concepts

Intervals

Intervals are a way of expressing a set of numbers between two points. They are incredibly useful in mathematics, especially when dealing with ranges of numbers.Different types of brackets are used to indicate whether endpoints are included.

• Round brackets, like \( ( \) or \( ) \), indicate that the endpoint is not included. This is known as an open interval.
• Square brackets, like \[ [ \text{or} ] \] , mean the endpoint is included, known as a closed interval.

The interval \((-\infty, 1)\) is open because it uses round brackets, excluding 1 from the set.Intervals can be finite or infinite. The infinite aspect here is shown by \(-\infty\), which denotes that the numbers can continue indefinitely in one direction. Hence, the phrase \'from negative infinity to 1\' describes a set of all numbers less than 1.

Number Line

A number line is a simple and vital tool for visualizing numbers. It helps us understand intervals and inequalities quickly.To draw a number line:

• Start with a horizontal line.
• Mark specific points, like zero and integers, to orient yourself.
• Locate and mark the specific value related to your interval. In our problem, this value is 1.

For the given interval \((-\infty, 1)\), you would:

• Place an open circle at 1 to indicate it is not included.
• Shade the line to the left of 1, representing all numbers less than 1.

The number line provides a visual representation of our interval, which can make comprehension of abstract concepts clearer.

Mathematical Notation

Mathematical notation is a system of symbols used to write mathematical concepts and relationships in a clear and concise way.In our exploration of the interval \((-\infty, 1)\), we have encountered several important symbols:

• The round brackets \( ( ) \) specify open intervals where the boundary points are not part of the interval.
• The infinity symbol \(\infty \) is used to indicate continuation indefinitely in mathematics. Negative infinity goes in the opposite direction.
• Inequality symbols, such as \( < \), express relationships between numbers. The notation \( x < 1 \) tells us all the numbers of \( x \) are less than 1.

Using this concise notation method, complex ideas become more manageable and communicable for mathematicians globally.