Problem 42 - Precalculus

Question

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

Step-by-Step Solution

Answer

The interval \((-2, 2)\) in inequality notation is \(-2 < x < 2\).

Step 1: Understand the Set Notation

The given interval \((-2,2)\) is in interval notation. This represents all real numbers \(x\) such that \(-2 < x < 2\). The parentheses \(()\) indicate that -2 and 2 are not included in the interval.

Step 2: Convert to Inequality Notation

To convert the interval \((-2,2)\) into inequality notation, we write the compound inequality: \(-2 < x < 2\). This means \(x\) is greater than -2 and less than 2, but not equal to -2 or 2.

Step 3: Sketch the Number Line Representation

To represent the interval on a number line, we draw a horizontal line. Place open circles at -2 and 2 to indicate these points are not included in the interval. Shade the region between -2 and 2 to denote all numbers between these points are included in the interval.

Key Concepts

Interval Notation

Interval notation is a mathematical way to describe a range of values. In general, it is expressed by two endpoints, which define the interval's beginning and end. The notation uses either parentheses or square brackets to specify whether the endpoints are included or not:

• Parentheses \( () \) indicate that the endpoints are not included, called an open interval.
• Square brackets \[ [] \] imply that the endpoints are included, referred to as a closed interval.

For example, the interval \((-2, 2)\) specifies all numbers between -2 and 2, excluding -2 and 2 themselves. This is an open interval. When dealing with open intervals, always remember that the boundaries are not part of the set. Using interval notation helps to concisely express a set of values, especially when stating output or solution sets for functions, inequalities, or within calculus. In this format, it's clear and precise, eliminating the need for lengthy descriptions.

Inequality Representation

Inequality representation is another way to express the range of values described by an interval. When transforming an interval, like \((-2, 2)\), into inequality notation, we weaken our language to say what numbers are greater or less than specific points:
For \((-2, 2)\), the inequality notation is \(-2 < x < 2\). This compound inequality tells us several things:

• \