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

Verified step by step guidance

1. Understand that the absolute value inequality \(|x| < 3\) means the distance of \(x\) from 0 on the number line is less than 3.
2. Rewrite the inequality without the absolute value by expressing it as a compound inequality: \(-3 < x < 3\).
3. Interpret this compound inequality as all values of \(x\) that lie strictly between -3 and 3.
4. Express the solution set in interval notation as \((-3, 3)\), indicating all numbers between -3 and 3 but not including -3 and 3 themselves.
5. Verify your solution by testing values inside the interval (like 0 or 2) and outside the interval (like -4 or 4) to confirm the inequality holds true only within the interval.

Key Concepts

Absolute Value Definition

The absolute value of a number represents its distance from zero on the number line, always as a non-negative value. For any real number x, |x| equals x if x is positive or zero, and -x if x is negative. Understanding this helps interpret inequalities involving absolute values.

Solving Absolute Value Inequalities

An inequality like |x| < a, where a is positive, means the distance of x from zero is less than a. This can be rewritten as a compound inequality: -a < x < a. Recognizing this equivalence is essential for solving absolute value inequalities.

Number Line Interpretation

Visualizing absolute value inequalities on a number line aids comprehension. For |x| < 3, the solution includes all points within 3 units of zero, i.e., between -3 and 3. This graphical approach helps confirm and understand the solution set.