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Chapter 4: Problem 108

Write an equivalent inequality using absolute value. $$ x<-4 \text { or } 4

Short Answer

Expert verified
\( |x| > 4 \)

Step by step solution

01

Analyze the Inequality

The given inequality is two separate conditions: 1. \( x < -4 \) 2. \( x > 4 \)Both conditions can be represented using a single absolute value inequality.
02

Understand the Absolute Value

The absolute value inequality \(|x| > a\) means that the distance of \(x\) from 0 is greater than \(a\). This could be interpreted as \( x > a \) or \( x < -a \).
03

Apply Absolute Value to the Given Inequality

For the given inequality, we need to find the value of \(a\). Notice that \( x < -4 \) or \( x > 4 \) can be written as the absolute value inequality \( |x| > 4 \) since if the absolute value of \(x\) is greater than 4, \(x\) must be less than -4 or greater than 4.
04

Write the Final Answer

The equivalent absolute value inequality for the given conditions is \( |x| > 4 \).

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Key Concepts

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

Inequality
An inequality is a mathematical statement that compares two values, expressions, or quantities which may not necessarily be equal. It uses symbols like <, >, 鈮�, and 鈮�. Inequalities show if one side is smaller, greater, or potentially equal to the other side. Unlike equations, which denote exact equality, inequalities highlight relational disparity under certain conditions. Splitting inequalities into ranges or using tricks like absolute values can simplify understanding.
Absolute Value
The absolute value of a number is its distance from zero on the number line, irrespective of direction. Mathematically speaking, the absolute value of a number x is written as |x|. For any real number, absolute value makes the number non-negative:
  • If x is positive or zero, \(|x| = x\).
  • If x is negative, \(|x| = -x\).
The concept is essential in inequalities, as it translates non-directional 'distance' into understandable ranges.
Distance from Zero
Distance from zero directly relates to the absolute value concept. Imagine placing a number on a number line; how far it is from the zero point reflects its absolute value. E.g., both 4 and -4 are four units away from zero, hence \(|4| = 4\) and \(|-4| = 4\). In the context of inequalities, this distance helps determine if numbers belong within specific ranges or exceed certain bounds. This understanding simplifies solving and understanding inequalities.
Equivalent Expressions
Equivalent expressions are different expressions that denote the same value. In the context of inequalities and absolute values, finding equivalent expressions allows us to simplify or restate a problem clearly. As seen in the exercise:
  • The inequality \( x < -4 \) or \( x > 4 \)
  • can be restated in the equivalent form \( |x| > 4 \)
This transformation helps to visualize and solve problems more efficiently.

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