In Exercises 59–94, solve each absolute value inequality. |3 - (2/3)x| > 5

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 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 the two cases that arise from the definition of absolute value, leading to two separate inequalities to solve.

Solving Linear Inequalities

Solving linear inequalities involves finding the values of the variable that satisfy the inequality. This process often includes isolating the variable on one side of the inequality sign. When dealing with absolute value inequalities, after breaking them into two cases, one must solve each resulting linear inequality and then combine the solutions to find the overall solution set.