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Chapter 1: Problem 14

Solve the equations. a. \(|w|=2\) b. \(|w|=0\) c. \(|w|=-2\)

Short Answer

Expert verified
a. \(w = 2 \text{ or } -2\)b. \(w = 0\)c. No solution

Step by step solution

01

Understanding absolute value

The absolute value of a number, written \(|n|\), represents the distance of the number from 0 on the number line. It is always non-negative.
02

Step 1a: Solve \(|w| = 2\)

Since the absolute value of a number equals 2, \(|w| = 2\) implies that \(w\) could be either 2 or -2, because both \(|2| = 2\) and \(|-2| = 2\).
03

Step 1b: Solve \(|w| = 0\)

The absolute value of a number equals 0 only if the number itself is 0. Therefore, \(|w| = 0\) implies \(w = 0\).
04

Step 1c: Solve \(|w| = -2\)

Since the absolute value is always non-negative, there is no number \(w\) such that \(|w| = -2\). Thus, there is no solution for this equation.

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Key Concepts

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

absolute value properties
Absolute value is a fundamental concept in mathematics that measures the distance of a number from zero on the number line, regardless of direction. This distance is always non-negative. For any real number \(n\), its absolute value is denoted as \(|n|\). Some important properties of absolute values include:
  • \(|n| \geq 0\) for all real numbers \(n\): The absolute value is always zero or positive.
  • \(|-n| = |n|\): The absolute value of a number is the same as the absolute value of its negative.
  • \(|0| = 0\): The absolute value of zero is zero.
These properties make the absolute value a robust tool in equations and inequalities.
solving absolute value equations
Solving absolute value equations involves finding all possible values that satisfy the equation. Here's a rundown of how to do that:
  • With \(|w| = 2\), we note that the absolute value expression gives us two potential solutions. This means \(w\) could be either 2 or -2 because both \(|2| = 2\) and \(|-2| = 2\).
  • When dealing with \(|w| = 0\), we realize that the absolute value of zero is zero. Thus, the only solution here is \(w = 0\).
  • For any equation like \(|w| = -2\) where the right-hand side is negative, there is no solution. Since absolute values can never be negative, no value of \(w\) will satisfy the equation.
By understanding these methods, you can solve any absolute value equation methodically and correctly.
non-negative values
A non-negative value is any real number that is either positive or zero. Absolute values naturally fall into this category because they represent distances.
Here's how non-negative values play a role in absolute value equations:
  • Absolute values measure how far a number is from zero, disregarding its direction. This means they can never be negative.
  • When solving equations like \(|w| = 2\), the solutions are the numbers that are exactly 2 units away from zero, which can be 2 or -2.
  • For \(|w| = 0\), the only number that is zero units away from zero is zero itself, hence \(w = 0\).
  • Equation \(|w| = -2\) immediately tells us there are no solutions since absolute values cannot be negative.
Recognizing the significance of non-negative values helps reinforce why absolute value problems behave the way they do.

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