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Chapter 1: Problem 97

Solve. Consider only integer replacements. $$|x|<3$$

Short Answer

Expert verified
-2, -1, 0, 1, 2

Step by step solution

01

Understand Absolute Value Inequality

The inequality is given as \( |x| < 3 \). This means that the absolute value of x is less than 3.
02

Interpret the Inequality

The absolute value \( |x| \) represents the distance of x from 0. For \( |x| < 3 \), x must be within 3 units from 0. This can be written as two separate inequalities: \(-3 < x < 3\).
03

Determine Integer Solutions

Since only integer replacements are considered, identify all the integers that satisfy \(-3 < x < 3\). These integers are \(-2, -1, 0, 1, \text{and} 2\).

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Key Concepts

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

Integer Solutions
When solving absolute value inequalities that demand integer solutions, it is important to think about whole numbers. Unlike real numbers, integers do not include fractions or decimals. They are simply the counting numbers (both positive and negative), including zero. In the exercise, the inequality given is \(|x| < 3\). We are looking for all the integers (whole numbers) that lie within this range.
Inequalities
In mathematics, inequalities represent the relationship between two values potentially being different. They can show that one value is smaller, larger, or within a certain range relative to another value. There are a few different symbols used in inequalities:
  • \(<\) means 'less than'
  • \(>\) means 'greater than'
  • \(\leq\) means 'less than or equal to'
  • \(\geq\) means 'greater than or equal to'
For absolute value inequalities like \(|x| < 3\), it leads to a compound inequality which translates to \(-3 < x < 3\). This means x has to be greater than -3 and less than 3 simultaneously.
Absolute Value
The absolute value of a number is its distance from zero on the number line, regardless of direction. Absolute value is always a non-negative number. For example, \(|-3| = 3\) and \(|3| = 3\).In this exercise, \(|x| < 3\) means we want x to be less than 3 units away from zero. Mathematically, it is written as \(-3 < x < 3\). Moving forward, we then find the integer values within this interval: -2, -1, 0, 1, and 2.

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