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Chapter 8: Problem 118

Find an equivalent inequality with absolute value. $$ -5

Short Answer

Expert verified
\(|x + 2| < 3\)

Step by step solution

01

Identify the Middle Point

Identify the central point between the endpoints of the inequality. In this case, the middle point is \(\frac{-5 + 1}{2} = -2\).
02

Determine the Distance from the Midpoint

Calculate the distance from the midpoint to either endpoint. The distance is \(|1 - (-2)| = 3\) or \(|-2 - (-5)| = 3\).
03

Set Up the Absolute Value Inequality

Express the given inequality in terms of the midpoint and the distance calculated. This can be written as \(|x + 2| < 3\).

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Key Concepts

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

inequalities
In mathematics, inequalities help us understand the range of values that a variable can take. For example, when we say \(-5 < x < 1\), we mean that \(x\) can be any number between \(-5\) and \(1\), but not including \(-5\) or \(1\).

There are different types of inequalities such as strict inequalities (like our example) and non-strict inequalities(like \(-5 \leq x \leq 1\)), which include the endpoints.

Understanding inequalities is crucial because they are useful in a wide array of practical problems, from physics to economics.
absolute value
Absolute value deals with the distance of a number from zero on a number line, regardless of direction. It is always non-negative.

For a number \(a\), its absolute value is denoted as \(|a|\). For example, \(|-3| = 3\) and \(|4| = 4\).

In equations and inequalities, absolute value can represent a range of solutions. For instance, \(|x| < 3\) means that \(-3 < x < 3\). This is handy when dealing with deviations from a central point, just like in the given problem.

To express a range using absolute value, we often find the midpoint and the maximum distance from this midpoint to the endpoints.
midpoints
Midpoints are central to understanding the absolute value inequality. The midpoint is the average of the endpoints of a range.

For the inequality \(-5 < x < 1\), find the midpoint by adding the endpoints and dividing by two: \(\frac{-5 + 1}{2} = -2\).

Next, find the distance from this midpoint to either endpoint. This distance helps create an absolute value inequality that summarizes the range.

For our example, the distance is \(|1 - (-2)| = 3\) or \(|-2 - (-5)| = 3\).

Now you can write the range as an absolute value inequality: \(|x + 2| < 3\).This tells us that \(x\) is within 3 units of \(-2\), effectively summarizing the original inequality.

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