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The concept of absolute value is pivotal when working with numerical expressions, especially in the context of inequalities. At its core, the absolute value of a number represents its distance from zero on a number line, regardless of direction. Hence, it's always a non-negative number. For any real number 'a', the absolute value is denoted as \(|a|\) and is defined by:

\[|a| = \begin{cases} a & \text{if } a \geq 0, \ -a & \text{if } a < 0. \end{cases}\]
For instance, \(|-3| = 3\) and \(|4| = 4\), since both 3 and 4 are three and four units away from zero on the number line. So, when you see the absolute value symbol, remember that it's asking for the distance, not necessarily the original value of the number inside.

Solving an inequality means finding all the values that make the inequality true. It's similar to solving equations, but instead of equalities, we're looking for ranges of values that satisfy a 'greater than' (>) or 'less than' (<) relationship. When the inequality involves an absolute value, like \(|x-2|>9\), it represents a two-part question: one for when the quantity inside is positive, and one for when it's negative. You would solve the inequality \(|x-2|>9\) by considering the two scenarios:\[x - 2 > 9\] and \[x - 2 < -9\].

These two separate inequalities represent the distances on either side of zero: more than 9 units away, or less than -9 units away. The solution to an absolute value inequality, therefore, is typically a range of values, or sometimes two ranges, that do not include the points between which the absolute value lies.

When validating whether a specific number is a solution to an inequality, we use a process known as 'substitution'. This method involves replacing the variable with the given number and calculating the outcome to see if the inequality holds true. For example, to check if 7 is a solution for \(|x-2|>9\), we substitute 'x' with 7, yielding \(|7 - 2| > 9\). Upon simplifying \(|5| > 9\), we evaluate this to 5 > 9, which is false. Therefore, 7 is not a solution. Substituting values is a quick and effective way to check possible solutions or to evaluate expressions at particular points, and it's a fundamental skill for tackling a wide array of mathematical problems.