Problem 52 - Precalculus

Question

Express each interval using inequality notation and show the given interval on a number line. $$(-\infty,-2)$$

Step-by-Step Solution

Answer

The inequality notation is \(x < -2\) and it is shown on a number line with an open circle at \(-2\) and shading to the left.

Step 1: Understand the Interval Notation

The interval \((-\infty, -2)\) represents all real numbers less than \(-2\). The parenthesis indicates that \(-2\) is not included in the interval.

Step 2: Convert to Inequality Notation

Since the interval includes all numbers less than \(-2\) and not including \(-2\), it can be written in inequality notation as \(x < -2\).

Step 3: Draw the Number Line

Create a horizontal line and select a point to represent \(-2\).

Step 4: Illustrate the Interval on the Line

Draw an open circle at \(-2\) to indicate that \(-2\) is not included. Shade the line to the left of \(-2\) to show that numbers less than \(-2\) are included in the interval.

Key Concepts

Inequality Notation

Inequality notation is a way to express a range or set of numbers that meet specific conditions. It uses symbols such as ">", "<", "≥", and "≤" to denote the relationship between numbers.
In this exercise, the inequality notation used is "\(x < -2\)", indicating that we're interested in all numbers that are less than \(-2\). It's important to note that the inequality does not include the number \(-2\) itself, since we use "<" instead of "≤". This open nature of the inequality means only numbers strictly less than \(-2\) satisfy it.
Inequality notation is valuable because it provides a clear and concise way to describe a set of numbers. It allows us to avoid listing an infinite number of elements explicitly when they share a common trait. Remembering these key points will help you master how to write and interpret inequalities effectively.

Number Line

A number line is a visual representation of numbers on a straight line. This tool helps to visualize intervals and how they relate to specific values. To depict the interval \((-\infty, -2)\) on a number line, start by drawing a straight horizontal line.
Choose a point on this line to represent \(-2\). Since \(-2\) is not included in the interval, you'll place an open circle at this point. The open circle signifies that the boundary does not include the number itself. This is an important concept as it differentiates open intervals from closed ones, where the boundary number would be part of the set.
Next, shade the portion of the line to the left of \(-2\). This shading represents all real numbers less than \(-2\), extending indefinitely towards negative infinity. Using a number line effectively bridges abstract inequality notation and visual understanding.

Real Numbers

Real numbers are an essential part of mathematics, encompassing all the rational numbers, such as integers and fractions, and all the irrational numbers, like \(\pi\) and the square root of \(-1\). Real numbers can be positive, negative, or zero, and they can also be whole numbers or decimals.
In the context of the interval \((-\infty, -2)\), when stating that the interval includes all real numbers less than \-2\, it implies every possible number on the number line that is to the left of \-2\. This coverage includes both rational numbers, like \(-3\) and \(-2.5\), and irrational numbers like \(-\sqrt{10}\).
Real numbers provide a comprehensive framework for discussing intervals and inequalities. Understanding this helps in interpreting intervals like \((-\infty, -2)\) correctly, ensuring that all relevant numbers are considered in the mathematical solution.