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Chapter 1: Problem 69

In Exercises 59鈥�94, solve each absolute value inequality. $$ |x|>3 $$

Short Answer

Expert verified
The solution to the inequality \(|x|>3\) is \(x>3\) or \(x<-3\). This means the values of \(x\) that satisfy the inequality are all real numbers greater than 3, and all real numbers less than -3.

Step by step solution

01

Understand absolute values

An absolute value of a real number \(x\), denoted \(|x|\), is the distance of \(x\) from zero. In context of the given inequality \(|x|>3\), it represents all points on the number line that are more than 3 units from zero. That includes points to the right of 3 and to the left of -3.
02

Formulate the inequality considering absolute values

The inequality \(|x|>3\), can be split into two separate inequalities. If \(x\) is the the distance from zero, then the values greater than 3 could be on the positive side (right side of number line, where \(x>3\)) or on the negative side (left side of number line, where \(x<-3\)).
03

Combine the inequalities

Therefore, solving the inequality \(|x|>3\) leads to finding the numbers \(x\) such that \(x>3\) or \(x<-3\) which are two distinct ranges on the number line.

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Key Concepts

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

Real Numbers
Real numbers are a fundamental part of mathematics and are the building blocks for understanding many mathematical concepts, including inequalities. Real numbers include all the numbers that we can find on the number line. This encompasses a wide variety of numbers, such as:
  • Whole numbers (e.g., 0, 1, 2)
  • Integers (e.g., -3, -2, 1)
  • Rational numbers (fractions like 1/2 and decimals like 0.5)
  • Irrational numbers (such as \(\pi\) and \(\sqrt{2}\))
Real numbers are crucial in solving inequalities because they provide a comprehensive set of values from which solutions can be drawn. When dealing with absolute value inequalities, like \(|x| > 3\), we understand that \(x\) can take any real number greater than 3 or less than -3.
Number Line
A number line is a visual representation of all real numbers. It stretches indefinitely in both directions, with zero at the center. Using a number line helps us visualize solutions to equations and inequalities effectively.

When solving absolute value inequalities such as \(|x| > 3\), the number line gives us a clear view of where the solutions lie:
  • Numbers greater than 3 are found on the right side of zero.
  • Numbers less than -3 are found on the left side of zero.
Understanding that the number line is a straight way to find all possible solution ranges helps keep things clear and straightforward. It's especially useful in identifying solutions when dealing with absolute values, as it spans negative and positive directions around zero.
Inequality Solution
An inequality solution involves finding the set of all values that satisfy the inequality condition. For the absolute value inequality \(|x| > 3\), our task is to find values of \(x\) that are more than 3 units from zero on the number line.Breaking it down:
  • Inequality \(|x| > 3\) splits into two parts: \(x > 3\) and \(x < -3\).
  • These inequalities indicate that solutions can exist in two separate ranges.
  • The solution set is \(( -\infty, -3 ) \cup ( 3, \infty)\). This tells us that any number less than -3 or greater than 3 satisfies the inequality.
By understanding the inequality solution in terms of real numbers, we map out these solutions on our number line, making it easier to grasp the concept effectively.
Inequality Formulation
Inequality formulation is the process of creating an inequality statement that represents a given condition or rule, such as an absolute value inequality like \(|x| > 3\). Formulating an inequality requires an understanding of how to split and manipulate the absolute value into understandable and solvable parts.Here's how you formulate the absolute value inequality:
  • Start with \(|x| > 3\).
  • Recognize that absolute value represents the distance from zero, hence, there are two potential conditions:
  • \(x > 3\), indicating distance greater than 3 in the positive direction.
  • \(x < -3\), indicating distance greater than 3 in the negative direction.
Formulating inequalities correctly allows you to determine all potential solutions by creating separate, simple inequalities that are easy to solve. This also aids in visualizing these solutions on the number line or in real-world problem-solving scenarios.

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