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In mathematics, inequalities express the relationship between two expressions that are not necessarily equal. They reveal the difference in value through symbols like greater than (>) and less than (<). Inequalities demonstrate how quantities relate and order themselves.
When we talk about absolute value inequalities, as in this case of \(|8x+3| > 12\), we essentially handle inequalities involving absolute values.This indicates the number's distance from zero on the number line.
Notice that absolute value inequalities can result in two separate expressions. For instance, \(|A| > B\) becomes two cases: \(A > B\) and \(A < -B\). These scenarios cover the full range of numbers exceeding a certain distance from zero.

Interval notation is a concise way to express set of solutions for inequalities. It's a shorthand method to describe intervals on a number line.
For example, in our solution \(x > \frac{9}{8}\) or \(x < -\frac{15}{8}\), the solution set is represented as \((-fty, -\frac{15}{8}) \cup (\frac{9}{8}, \infty)\). Here, each part of the interval uses parentheses \(()\) to show that endpoints are not included in the solution, known as open intervals.
The union symbol \(\cup\) indicates that the solutions for both separate conditions are combined. This way, interval notation efficiently handles the complex task of writing solutions without lengthy expressions.

Graphing inequalities provides a visual representation of solution sets on a number line.
In this case, we have two open intervals: one to the left of \(-\frac{15}{8}\) and another to the right of \(\frac{9}{8}\).To graph these, place an open circle at each endpoint and shade the entire region stretching to infinity on either side.
Open circles denote that endpoints are not part of the solution, keeping the intervals open.This graph vividly shows all values \(x\) that satisfy the inequality \(|8x+3| > 12\), giving a clear and immediate understanding of the solution set. Additionally, graphing such inequalities aids in connecting algebraic expressions with their geometric representations.