In Exercises 59–94, solve each absolute value inequality. |x| < 3

Verified step by step guidance

1. Understand that the inequality \(|x| < 3\) means that the distance of \(x\) from 0 on the number line is less than 3.
2. Rewrite the absolute value inequality \(|x| < 3\) as a compound inequality: \(-3 < x < 3\).
3. This compound inequality means that \(x\) is greater than \(-3\) and less than \(3\).
4. Express the solution set in interval notation: \((-3, 3)\).
5. Verify the solution by checking that any number within the interval \((-3, 3)\) satisfies the original inequality \(|x| < 3\).

Key Concepts

Absolute Value

Absolute value measures the distance of a number from zero on the number line, regardless of direction. For any real number x, the absolute value is denoted as |x| and is defined as |x| = x if x ≥ 0, and |x| = -x if x < 0. Understanding absolute value is crucial for solving inequalities that involve it, as it transforms the problem into a comparison of distances.

Inequalities

Inequalities express a relationship between two values that are not necessarily equal, using symbols such as <, >, ≤, or ≥. In the context of absolute value inequalities, they indicate the range of values that satisfy the condition. For example, |x| < 3 means that x is within 3 units of 0, leading to a solution that encompasses a range of values rather than a single point.

Solving Absolute Value Inequalities

To solve an absolute value inequality like |x| < 3, one must break it down into two separate inequalities: -3 < x < 3. This process involves understanding that the absolute value creates two scenarios: one where the expression inside is positive and one where it is negative. The solution set is then derived from these inequalities, providing the range of values that satisfy the original condition.