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Chapter 4: Problem 109

Write an equivalent inequality using absolute value. $$ x<-8 \text { or } 2

Short Answer

Expert verified
The equivalent inequality in absolute value form is \( |x| > 8 \).

Step by step solution

01

Understanding the inequality

We start with the compound inequality: \[ x < -8 \text{ or } x > 2 \]
02

Express as two separate inequalities

Rewrite the compound inequality as two separate inequalities: 1. \( x < -8 \)2. \( x > 2 \)
03

Determine the distance from the midpoint

Notice that both inequalities describe values of \( x \) being further than a certain distance from the midpoint between -8 and 2. The midpoint is 0, so we find the distance from 0 to each boundary. The distances are: \[ |-8 - 0| = 8 \]\[ |2 - 0| = 2 \]
04

Choose the larger distance

The larger distance that defines our inequality is 8. Therefore, we express \( x \) being further than 8 units from 0. In absolute value terms, this is: \[ |x| > 8 \]

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Key Concepts

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

compound inequalities
Compound inequalities involve two separate inequalities that are combined by the words 'and' or 'or'. When we say \( x < -8 \text{ or } \) it means that we are looking for values of \(x\) that satisfy either one or both of the inequalities.

In the given exercise, we have:
  • \( x < -8 \)
  • \( x > 2 \)

This means we are interested in values of \(x\) that are either less than \(-8\) or greater than \(2\). Each inequality describes a separate range of values for \(x\). Knowing how to solve compound inequalities is important because it helps break down complex ranges of solutions into more manageable pieces.
midpoint
The midpoint is the point that is exactly halfway between two numbers. To find the midpoint between \(-8\) and \(2\), we use the formula:

\[ \text{Midpoint} = \frac{a + b}{2} \]

Where \(a\) and \(b\) are the given numbers.

Let's calculate it:

\[ \text{Midpoint} = \frac{-8 + 2}{2} = \frac{-6}{2} = -3 \]

However, this value isn't used directly in the given problem. Instead, we recognize that the midpoint we are focusing on for our more intuitive understanding is actually \(0\), which is not the algebraic but rather a contextual midpoint given the exercise. This helps simplify the concept and stay aligned with deriving the absolute value inequality.
distance in inequalities
Distance helps to convert a compound inequality into an absolute value inequality. We first need to determine the distance of each boundary value from the midpoint \(0\).

Let's compute these distances:

  • Distance from -8 to 0 is \(|-8 - 0| = 8\)
  • Distance from 2 to 0 is \(|2 - 0| = 2\)


The larger distance, 8, is what we use to express the absolute value inequality:
\[ |x| > 8 \]
This tells us that the values of \(x\) that satisfy this inequality are further away from \(0\) than \(8\) units. Thus, values for \(x\) there are less than \( -8\) or greater than \(2\).

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