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

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Understand that the absolute value inequality $|x| < 3$ means the distance of $x$ from 0 on the number line is less than 3.

Rewrite the inequality without the absolute value by expressing it as a compound inequality: $-3 < x < 3$.

Interpret this compound inequality as all values of $x$ that lie strictly between -3 and 3.

Express the solution set in interval notation as $(-3, 3)$, indicating all numbers between -3 and 3 but not including -3 and 3 themselves.

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.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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