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Chapter 4: Problem 107

Write an equivalent inequality using absolute value. $$ x \leq-6 \text { or } 6 \leq x $$

Short Answer

Expert verified
\( |x| \geq 6 \)

Step by step solution

01

Understand the Given Inequality

The inequality given is a compound inequality: \( x \leq -6 \text{ or } 6 \leq x \). This means the values of \( x \) are either less than or equal to -6, or greater than or equal to 6.
02

Rewrite the Inequality in Absolute Value Terms

The given compound inequality can be rewritten using absolute value. The absolute value measures the distance of a number from 0. Notice that the distances from 0 to -6 and from 0 to 6 are both 6.
03

Express the Condition with Absolute Value

Numbers that are either less than or equal to -6 or greater than or equal to 6 can be written as: \( |x| \geq 6 \). This shows that the distance of \( x \) from 0 is at least 6.

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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 combined by 'and' or 'or'. In our problem, the inequality is given as \( x \leq -6 \text{ or } 6 \leq x \). This means we are looking for all values of \( x \) that are either less than or equal to -6, or greater than or equal to 6.

Let's break it down:
  • With \( x \leq -6 \), we consider all numbers smaller than or equal to -6.
  • With \( 6 \leq x \), we consider all numbers greater than or equal to 6.
The 'or' in this context means that \( x \) can satisfy either of these conditions. This is fundamentally different from a compound inequality using 'and', where \( x \) would need to satisfy both conditions simultaneously.
Absolute Value
Absolute value represents the distance of a number from zero on the number line, regardless of direction. For example, both -3 and 3 have an absolute value of 3 because they are both 3 units away from zero.

In mathematical terms, the absolute value of \( x \) is written as \( |x| \). It essentially strips away any negative sign. So, \( |-6| = 6 \) and \( |6| = 6 \).

Applying this to our inequality, we notice that 6 is the distance from zero for both -6 and 6. So, when we look at \( x \leq -6 \text{ or } 6 \leq x \), we are really looking at numbers that are at least 6 units away from zero.
Inequality Transformation
Transforming the given compound inequality \( x \leq -6 \text{ or } 6 \leq x \) into an absolute value inequality helps to unify the expression and simplify the understanding.

Here's how we do it:
  • Recognize that the distance condition stated can be described using absolute values.
  • Numbers that are at least 6 units away from zero can be written as \( |x| \geq 6 \).
This transformation shows that the quantity we're interested in is the distance of \( x \) from zero being at least 6 units. The key here is understanding that \( |x| = x \) if \( x \geq 0 \) and \( |x| = -x \) if \( x < 0 \). Thus, \( x \leq -6 \text{ or } 6 \leq x \) becomes \( |x| \geq 6 \), which covers both ranges in one concise statement.

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Most popular questions from this chapter

Solve and graph. Write the answer using both set-builder notation and interval notation. Let \(f(x)=|5 x+2| .\) Find all \(x\) for which \(f(x) \leq 3\)

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The formula $$ C=\frac{5}{9}(F-32) $$ is used to convert Fahrenheit temperatures \(F\) to Celsius temperatures \(C\). a) Gold is liquid for Celsius temperatures \(C\) such that \(1063^{\circ} \leq C<2660^{\circ} .\) Find a comparable inequality for Fahrenheit temperatures. b) Silver is liquid for Celsius temperatures \(C\) such that \(960.8^{\circ} \leq C<2180^{\circ} .\) Find a comparable inequality for Fahrenheit temperatures.

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Solve and graph. Write the answer using both set-builder notation and interval notation. $$ \left|\frac{1+3 x}{5}\right|>\frac{7}{8} $$
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