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Chapter 3: Problem 379

In the following exercises, evaluate. $$ |x| \text { when } x=-14 $$

Short Answer

Expert verified
|-14| = 14

Step by step solution

01

Understand Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is always non-negative.
02

Apply the Absolute Value Definition

The absolute value of a number is the number itself if it is positive or zero, and the opposite of the number if it is negative. So for any number x, \(|x| = x \text{ if } x \geq 0\) and \(|x| = -x \text{ if } x < 0\).
03

Evaluate the Given Expression

In this exercise, x is given as -14. Since -14 is less than 0, use the definition for negative numbers: \(|x| = -(-14)\).
04

Simplify the Expression

Simplify the expression: \(-(-14) = 14\).
05

Write the Final Answer

The absolute value of -14 is 14.

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Key Concepts

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

Absolute Value Definition
Absolute value is a fundamental concept in prealgebra. It measures how far a number is from zero on the number line. Think of it as the 'distance' from zero, without considering which direction you are moving.
For example:
  • The absolute value of 3 is 3, because 3 is three units away from zero.
  • The absolute value of -7 is 7, because -7 is seven units away from zero.
Unlike regular values, absolute values are never negative. The notation for absolute value is two vertical lines. So, if you see \(|x|\), it means the absolute value of x.
This concept is very useful when comparing distances or solving equations, as it simplifies working with negative numbers. Remember:
  • If x is positive or zero, \(|x| = x\).
  • If x is negative, \(|x| = -x\).
Negative Numbers
Negative numbers are values less than zero. They are often represented with a minus sign (-) in front. For example, -4, -10, and -50 are all negative numbers.
In many real-life situations, negative numbers represent loss or decrease. For example:
  • An elevation of -20 meters is 20 meters below sea level.
  • A temperature of -5 degrees means it's five degrees below zero.
Understanding negative numbers is crucial when dealing with absolute value. For negative numbers, the absolute value process involves taking the opposite of the number. This is because we want to find its 'distance' from zero. So, if x = -14:
  • The absolute value of -14 is \(|-14| = -(-14) = 14\).
Taking the opposite of a negative number turns it into its positive counterpart.
Evaluating Expressions
When evaluating an expression with absolute value, follow these key steps:
  • Identify if the number is positive, negative, or zero.
  • Apply the definition of absolute value.
  • Simplify the expression.
  • State the final answer.
Let's break these steps down using an example where x = -14:
1. Identify that -14 is a negative number.
2. Apply the absolute value definition for negative numbers: \(|x| = -x\).
3. Substitute -14 into the expression: \(|-14| = -(-14)\).
4. Simplify by removing the double negative: \(-(-14) = 14\).
5. Conclude that the absolute value of -14 is 14.
Remember, these steps help you work through an absolute value expression methodically. Practice more problems to solidify your understanding and always check your results!

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