In Exercises 59–94, solve each absolute value inequality.5|2x + 1| - 3 ≥ 9

Absolute value represents 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|, which equals x if x is non-negative and -x if x is negative. Understanding absolute value is crucial for solving inequalities that involve expressions within absolute value symbols.

Inequalities express a relationship between two expressions that are not necessarily equal. They can be represented using symbols such as >, <, ≥, or ≤. Solving inequalities often involves finding the range of values that satisfy the condition, which can include multiple solutions or intervals, especially when absolute values are involved.

To solve an absolute value inequality, one must consider the definition of absolute value and break the inequality into two separate cases. For example, the inequality |A| ≥ B leads to two scenarios: A ≥ B or A ≤ -B. This approach allows for finding all possible solutions that satisfy the original inequality, which is essential for complete problem-solving.