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Chapter 1: Problem 76

Express the fact that \(x\) differs from -4 by less than 1 as an inequality involving an absolute value. Solve for \(x\).

Short Answer

Expert verified
The solution is \(-5 < x < -3\).

Step by step solution

01

Understanding the Problem

Identify what it means for a number to differ from -4 by less than 1. This means the distance between the number and -4 is less than 1.
02

Set Up the Absolute Value Inequality

The distance between two numbers can be expressed using the absolute value function. The inequality can be written as: oThe absolute value of the difference between \(x\) and -4 should be less than 1.\[ |x + 4| < 1 \]
03

Solve the Inequality

To solve the inequality, split it into its two linear inequalities: \[ -1 < x + 4 < 1 \]
04

Isolate \(x\)

Subtract 4 from all parts of the compound inequality to solve for \(x\): \[ -1 - 4 < x + 4 - 4 < 1 - 4 \] \[ -5 < x < -3 \]

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Key Concepts

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

Understanding Inequality
An inequality is a mathematical statement that compares two expressions using inequality signs like <, >, 鈮�, or 鈮�. Instead of stating that two quantities are equal, inequalities show that one quantity is bigger, smaller, or different from another.
In this exercise, we start by saying that the difference in values is less than a specific number.
This gives us a way to understand how quantities relate to each other even when they are not identical.
Absolute Value Explained
The absolute value of a number represents its distance from zero on the number line, regardless of direction. For example, the absolute value of both 5 and -5 is 5, because both numbers are 5 units away from zero.
In this context, we use the absolute value to express the distance between two numbers.
This exercise uses the absolute value to show the distance between x and -4. When we say the distance is less than 1, we are really saying that \[ |x + 4| < 1 \].
Solving Inequalities
To solve the absolute value inequality, we first split it into two linear inequalities. This is because the absolute value inequality \[ |x + 4| < 1 \] can be understood as: \[ -1 < x + 4 < 1 \].
This approach breaks down the problem into manageable steps.
Next, isolate the variable x by applying the same mathematical operation to all parts of the compound inequality.
Compound Inequality
After splitting the absolute value inequality into two parts, we solve it as a compound inequality, meaning it has more than one inequality statement combined. For \[ -1 < x + 4 < 1 \], we solve for x by subtracting 4 from all parts:
\[ -1 - 4 < x + 4 - 4 < 1 - 4 \], which simplifies to \[ -5 < x < -3 \].
This tells us that solutions for x must be between -5 and -3. A compound inequality, therefore, provides a range of values that satisfy the inequality.

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Most popular questions from this chapter

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