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

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 leads to two separate cases based on the definition.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal, using symbols such as >, <, ≥, or ≤. In the context of absolute value inequalities, it is important to recognize that the absolute value can create two scenarios: one where the expression inside is greater than or equal to a positive number, and another where it is less than or equal to the negative of that number. This duality is essential for finding all possible solutions.

Solving Absolute Value Inequalities

To solve an absolute value inequality like |x - 1| ≥ 2, one must break it down into two separate inequalities: x - 1 ≥ 2 and x - 1 ≤ -2. This process involves isolating the variable in each case and then solving for x. The solutions from both inequalities are then combined to express the complete solution set, which may include intervals on the number line.