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Chapter 8: Problem 117

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

Short Answer

Expert verified
\( | x + 3 | > 5 \)

Step by step solution

01

- Understand the Given Inequalities

The problem consists of two inequalities: 1. \( x < -8 \) 2. \( x > 2 \). You need to find an equivalent single inequality involving absolute value.
02

- Identify the Distance from a Central Point

Notice that both inequalities deal with values either less than -8 or greater than 2. The midpoint between -8 and 2 is -3. Consider how far each boundary is from -3. \( -8 \) is \( 5 \) units away from -3 and \( 2 \) is also \( 5 \) units away from -3.
03

- Write the Absolute Value Inequality

Since any value satisfying either \( x < -8 \) or \( x > 2 \) will be at least 5 units away from -3, you can write the equivalent absolute value inequality as: \( | x + 3 | > 5 \).

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Key Concepts

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

inequalities
In mathematics, inequalities are statements about the relative size or order of two objects. They are crucial in algebra and help us determine ranges and constraints of values.
There are several types of inequalities, including:
  • Strict inequalities: these use symbols like < and > to imply one value is less than or greater than another without equality.
  • Non-strict inequalities: these use 鈮� and 鈮� to include the possibility of equality.
An example is the inequality \( x < -8 \) or \( x > 2 \) from our given problem. The challenge here is to combine these two separate inequalities into a single one that relates to the absolute value.
absolute value
The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always a non-negative number. For instance, \( |3| = 3 \) and \( |-3| = 3 \). This concept is useful in expressing distances or deviations from a central value.
In the given problem, we identify that values of \(x \) are either less than -8 or greater than 2. Simplifying this with absolute values, we calculate the midpoint between -8 and 2, which is \( -3 \). The distance from -3 to either boundary (-8 or 2) is 5 units. Thus, the inequality can be written as \( | x + 3 | > 5 \), indicating all values more than 5 units away from -3.
algebra
Algebra involves working with equations, inequalities, and expressions using symbols and letters to represent numbers. It allows us to solve problems and find unknown values.
When solving inequalities involving absolute values, it鈥檚 important to isolate the absolute value expression and consider the distance being represented. The inequality \( x < -8 \) or \( x > 2 \) can be transformed into an absolute value inequality, simplifying analysis and solution.
To summarize:
  • Identify the target intervals: \( x < -8 \) or \( x > 2 \)
  • Find the midpoint and the distance from it to boundary values.
  • Translate to absolute value form.
Here we get \( | x + 3 | > 5 \), encapsulating the values that satisfy either original inequality.

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