In Exercises 59–94, solve each absolute value inequality. |2(x - 1) + 4| ≤ 8

Key Concepts

Absolute Value

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

Solving Linear Inequalities

Solving linear inequalities involves isolating the variable on one side of the inequality sign. This process is similar to solving linear equations but requires careful attention to the direction of the inequality, especially when multiplying or dividing by negative numbers, which reverses the inequality sign. Mastery of this concept is essential for finding the solution set of the given absolute value inequality.