In Exercises 59–94, solve each absolute value inequality. |3x + 5| < 17

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 'a', the absolute value is denoted as |a| and is defined as |a| = a if a ≥ 0, and |a| = -a if a < 0. Understanding absolute value is crucial for solving inequalities that involve expressions within absolute value bars.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal. They can be strict (using < or >) or non-strict (using ≤ or ≥). When solving absolute value inequalities, it is important to translate the absolute value expression into two separate inequalities to find the solution set.

Solving Absolute Value Inequalities

To solve an absolute value inequality like |3x + 5| < 17, you split it into two cases: 3x + 5 < 17 and 3x + 5 > -17. This results in a compound inequality that can be solved to find the range of values for 'x' that satisfy the original inequality. Understanding how to manipulate and solve these cases is essential for finding the correct solution.