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Chapter 1: Problem 103

The equation \(|x|=-2\) has no solution. Explain why.

Short Answer

Expert verified
The equation has no solution because the absolute value can never be negative.

Step by step solution

01

Understand the Absolute Value Concept

Absolute value, represented by |x| , signifies the non-negative distance between a number x and zero on the number line. It means the result is always zero or positive.
02

Analyze the Given Equation

The equation given is |x| = -2 . Since |x| denotes the distance, it cannot be negative. As a result, -2 is not a valid value for |x|.
03

Conclude Solutions

Since the absolute value can never be negative, there are no values of x that satisfy the equation |x| = -2 .

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Key Concepts

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

Absolute Value: An Overview
Absolute value, noted as \(|x|\), is a fundamental mathematical concept that indicates the distance of a number from zero on the number line. This distance is always non-negative, meaning it is either positive or zero. Therefore, \(|3| = 3\) and \(|-3| = 3\). In both cases, the absolute value strips the number of its sign, focusing only on its magnitude. This concept is key in recognizing that an absolute value expression can never yield a negative result.
Equation Analysis: Understanding the Problem
In our given exercise, we analyze the equation \(|x| = -2\). Since absolute value represents the non-negative distance of a number from zero, it inherently cannot be negative. This analysis reveals the intrinsic nature of absolute value functions. When we equate \(|x|\) to a negative number, such as \(-2\), we recognize a fundamental inconsistency. This type of equation presents an impossibility in mathematics, as no real number exists whose absolute value equates to \(-2\).
Non-Negative Distance: Conceptual Understanding
The term 'non-negative distance' refers to measurements that are either positive or zero. Given that distance cannot be less than zero, absolute value functions adhere strictly to this principle. Consider the exercise's equation \(|x| = -2\). Here, the concept of non-negative distance ensures there are no real numbers that satisfy this condition. This leads to the conclusion that the equation has no solution. Such exercises help reinforce understanding that absolute values always produce non-negative outputs.

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Most popular questions from this chapter

IQ Tests A standard intelligence test has an average score of \(100 .\) According to statistical theory, of the people who take the test, the \(2.5 \%\) with the highest scores will have scores of more than \(1.96 \sigma\) above the average, where \(\sigma\) (sigma, a number called the standard deviation) depends on the nature of the test. If \(\sigma=12\) for this test and there is (in principle) no upper limit to the score possible on the test, write the interval of possible test scores of the people in the top \(2.5 \%\)

Multiple Choice A chemist mixes 10 liters of a \(20 \%\) solution with \(x\) liters of a \(35 \%\) solution. Which of the following expressions represents the total number of liters in the mixture? (a) \(x\) (b) \(20-x\) (c) \(\frac{35}{x}\) (d) \(10+x\)

Write each inequality using interval notation, and graph each inequality on the real mumber line. \(x \geq-3\)

Express the fact that \(x\) differs from -4 by less than 1 as an inequality involving an absolute value. Solve for \(x\).

The fact that \(2(x+3)=2 x+6\) is attributable to the _________ Property.
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