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Chapter 1: Problem 5

True or False The equation \(|x|=-2\) has no solution.

Short Answer

Expert verified
True

Step by step solution

01

Understanding Absolute Value

The absolute value of a number is its distance from zero on the number line. Absolute values are always non-negative. That is, for any real number x, the absolute value \(|x|\) is always greater than or equal to 0.
02

Analyzing the Equation

Given the equation \(|x| = -2\), recall that absolute values cannot be negative. This means there are no real numbers x that would satisfy this equation.
03

Concluding the Solution

Since the absolute value of a number cannot be -2, the equation \(|x| = -2\) has no solutions.

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Key Concepts

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

absolute value
The absolute value is a fundamental concept in mathematics. It represents the distance of a number from zero on the number line. Distance is always a non-negative quantity, so the absolute value of any real number is always greater than or equal to zero. For example, the absolute value of both -3 and 3 is 3, written as \(|3| = 3\) and \(|-3| = 3\). This means that regardless of whether a number is positive or negative, its absolute value is always expressed as a positive number or zero.
non-negative
Non-negative numbers are numbers that are either positive or zero. This distinction is important when discussing absolute values because it means that absolute values can never be negative. For instance, \(|5| = 5\) and \(|-5| = 5\). Both examples show that absolute values result in non-negative outcomes. In our exercise, we were given an equation \(|x| = -2\), which suggests that the absolute value of some number x is negative. This is impossible because, as we know, absolute values cannot be less than zero.
solution analysis
When solving equations involving absolute values, it's important to remember that you're working with a non-negative value. Consider the problem \(|x| = -2\). To find a solution, we need a number whose absolute value equals -2. But since absolute values cannot be negative, it's immediately clear that there are no real numbers x for which \(|x| = -2\). Therefore, the equation has no solutions. Solving absolute value equations involves analyzing the properties of absolute values carefully.
algebraic equations
Algebraic equations are mathematical statements that assert the equality of two expressions. In algebra, you solve equations to find the value of unknown variables that make the equation true. An absolute value equation like \(|x| = a\) can be solved by considering both cases where x could be either positive or negative. For example, \(|x| = 5\) means \(|x| = 5\) when x is 5 and \(|x| = -5\) when x is -5. However, if we encounter an equation like \(|x| = -2\), we must remember that there is no solution because an absolute value cannot be negative.

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Most popular questions from this chapter

In going from Chicago to Atlanta, a car averages 45 miles per hour, and in going from Atlanta to Miami, it averages 55 miles per hour. If Atlanta is halfway between Chicago and Miami, what is the average speed from Chicago to Miami? Discuss an intuitive solution. Write a paragraph defending your intuitive solution. Then solve the problem algebraically. Is your intuitive solution the same as the algebraic one? If not, find the flaw.

The distance to the surface of the water in a well can sometimes be found by dropping an object into the well and measuring the time elapsed until a sound is heard. If \(t_{1}\) is the time (measured in seconds) that it takes for the object to strike the water, then \(t_{1}\) will obey the equation \(s=16 t_{1}^{2}\), where \(s\) is the distance (measured in feet). It follows that \(t_{1}=\frac{\sqrt{s}}{4}\). Suppose that \(t_{2}\) is the time that it takes for the sound of the impact to reach your ears. Because sound waves are known to travel at a speed of approximately 1100 feet per second, the time \(t_{2}\) to travel the distance \(s\) will be \(t_{2}=\frac{s}{1100} .\) See the illustration. Now \(t_{1}+t_{2}\) is the total time that elapses from the moment that the object is dropped to the moment that a sound is heard. We have the equation $$ \text { Total time elapsed }=\frac{\sqrt{s}}{4}+\frac{s}{1100} $$ Find the distance to the water's surface if the total time elapsed from dropping a rock to hearing it hit water is 4 seconds.

Describe three ways that you might solve a quadratic equation. State your preferred method; explain why you chose it.

Find the real solutions, if any, of each equation. Use any method. $$ 2+z=6 z^{2} $$

Find the real solutions, if any, of each equation. $$ \left(\frac{y}{y-1}\right)^{2}=\frac{6 y}{y-1}+7 $$
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