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Chapter 4: Problem 112

Write an equivalent inequality using absolute value. \(x\) is less than 1 unit from 5

Short Answer

Expert verified
|x - 5| < 1

Step by step solution

01

Understand the Problem

Identify the key information given in the problem. The statement says that the distance between a number, denoted by \(x\), and the number 5 is less than 1 unit.
02

Set Up the Inequality

The absolute value expression can be used to represent the distance between two numbers. The inequality can be set up as follows: \[|x - 5| < 1\]This expression means the distance between \(x\) and 5 is less than 1.
03

Verify the Inequality

Consider that the absolute value \(|x - 5| < 1\) implies \[-1 < x - 5 < 1.\]
04

Solve the Double Inequality

To isolate \(x\), add 5 to all parts of the inequality: \[-1 + 5 < x - 5 + 5 < 1 + 5\]Simplify: \[4 < x < 6.\]

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Key Concepts

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

distance between numbers
Absolute value is a way to measure the distance between two numbers on a number line.
It always returns a non-negative value.
In this exercise, we are concerned with the distance between a number 饾懃 and the number 5 on the number line.
When we say '饾懃 is less than 1 unit from 5,' we essentially mean that the distance between 饾懃 and 5 is less than 1.
This can be mathematically written using absolute value as \(|x - 5| < 1\).
The absolute value symbol \( | \dots|\) ensures that we measure the distance regardless of the direction, meaning it doesn't matter if 饾懃 is to the left or right of 5.
inequality representation
Inequalities are mathematical expressions that show the relationship between two values where one is not necessarily equal to the other.
They use symbols like <, >, 鈮�, and 鈮�.
In our problem, \(|x - 5| < 1\) is an inequality that tells us the distance between 饾懃 and 5 is less than 1.
This can be understood by thinking in terms of a number line.
For example, if you imagine the number 5 on the number line, we're looking for all positions 饾懃 that are less than 1 unit away from 5.
This would be the range of values between 4 and 6, but not including 4 and 6.
solving double inequalities
A double inequality involves two inequality signs simultaneously.
The goal is to isolate the variable in the middle.
In our exercise, we start with: \(|x - 5| < 1\).
This absolute value inequality can be expanded to a double inequality: \(-1 < x - 5 < 1\).
To solve for 饾懃, we perform operations to isolate the variable.
In this instance, add 5 to all parts to get:
\(-1 + 5 < x - 5 + 5 < 1 + 5\)
This simplifies to: \(4 < x < 6\).
Therefore, 饾懃 can be any number between 4 and 6, but not equal to 4 or 6.
Understanding this process is crucial for solving more complex inequalities.

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