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Chapter 1: Problem 6

True or False The inequality \(|x| \geq-2\) has the set of real numbers as its solution set.

Short Answer

Expert verified
True.

Step by step solution

01

- Understanding the Absolute Value Function

First, consider the definition of absolute value. The absolute value function \(|x|\) represents the distance of a number \ x \ from zero on the number line, and this distance is always non-negative.
02

- Analyzing the Inequality

Look at the given inequality \(|x| \geq -2\). Given that \|x|\ is always non-negative, any non-negative number will be greater than or equal to -2. Therefore, this inequality is always true for any real number \ x \.
03

- Solution

Since the inequality \(|x| \geq - 2\) holds true for all real numbers \ x \, the solution set includes all real numbers \ x \.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Absolute Value
Let's start by looking at the concept of absolute value. Absolute value, denoted as \(|x|\), refers to how far a number is from zero on the number line. This measurement is always positive or zero鈥攊t never takes on a negative value.

For example, the absolute value of both -5 and 5 is 5, because both are five units away from zero.

Absolute value helps us understand the magnitude of a quantity regardless of its sign.
Understanding Inequality
Inequalities define a range of possible values a variable can take. In mathematical notation, inequality means 'greater than', 'less than', 'greater than or equal to', or 'less than or equal to'.

To make it simple, let's look at \(|x| \geq -2\). Normally, when dealing with inequalities, we consider the properties of both sides. Absolute value, as we discussed, is always non-negative. Hence, \(|x|\) can be 0 or any positive number.

Since 0 and positive numbers are always greater than or equal to \(-2\), this inequality will hold for any real number.
Solution Set of Inequality
Once we understand the inequality \(|x| \geq -2\), it becomes clear that any x satisfies this condition. Since absolute value is never negative, the 'solution set' here is pretty straightforward.

In this case, the solution set includes all real numbers because any real number, whether positive, negative, or zero, will always have an absolute value that is zero or positive, which is indeed greater than or equal to -2.

This means the solution set for the inequality \(|x| \geq -2\) is all real numbers, or mathematically, \(-\infty < x < \infty\).

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