In Exercises 59–94, solve each absolute value inequality. 12 < |- 2x + 6/7| + 3/7

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 defined as |x| = x if x ≥ 0 and |x| = -x if x < 0. In the context of inequalities, understanding how to manipulate absolute values is crucial for solving equations and inequalities that involve them.

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 consider both the positive and negative scenarios that arise from the definition of absolute value, leading to two separate cases to solve.

Isolating the Absolute Value

To solve an absolute value inequality, the first step is often to isolate the absolute value expression on one side of the inequality. This involves performing algebraic operations such as addition, subtraction, multiplication, or division. Once isolated, the inequality can be split into two separate inequalities, allowing for a systematic approach to find the solution set.