In Exercises 59–94, solve each absolute value inequality. - 2|x - 4| ≥ - 4

Key Concepts

Absolute Value

Absolute value represents the distance of a number from zero on the number line, regardless of direction. It is denoted by vertical bars, such as |x|, and is always non-negative. For example, |3| = 3 and |-3| = 3. Understanding absolute value is crucial for solving inequalities that involve expressions within these bars.

Inequalities

Inequalities express a relationship between two expressions that are not necessarily equal, using symbols like >, <, ≥, or ≤. In the context of absolute value inequalities, we often need to consider two cases: one for the positive scenario and one for the negative scenario. This duality is essential for finding all possible solutions to the inequality.

Solving Absolute Value Inequalities

To solve an absolute value inequality, we first isolate the absolute value expression and then set up two separate inequalities based on the definition of absolute value. For example, if |A| ≥ B, we solve A ≥ B and A ≤ -B. This method allows us to find all values of the variable that satisfy the original inequality, which is key to understanding the solution set.