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The absolute value of a number is its distance from zero on the number line, regardless of direction. Represented by two vertical bars on either side, like this: |x|, it converts all numbers into non-negative versions. For example, both |3| and |-3| equal 3 because both points are three units away from zero on the number line.

When dealing with absolute value inequalities such as |2x - 6| < 8, the inequality is essentially saying that the distance between '2x - 6' and '0' is less than 8 units on the number line. This concept is crucial for understanding how to split the problem into two parts: one for when (2x - 6) is positive and one for when it is negative, which leads us to consider different scenarios for the solutions.

Inequalities express the relationship between two values when they are not equal, using the symbols <, >, ≤, or ≥. Solving an inequality means finding all the possible values of a variable that make the inequality true.

With absolute value inequalities, you need to pay special attention to the direction of the inequality sign. When splitting the absolute value inequality |2x - 6| < 8 into two parts, each part retains the inequality from the original equation but in opposite directions, one with < and the other with >, forming a solution set that is a range of values satisfying both conditions.

Algebraic solutions involve manipulating equations or inequalities to solve for an unknown variable. The process usually includes operations such as adding, subtracting, multiplying, and dividing both sides of the equation to isolate the variable.

In the example provided, solving the inequalities 2x - 6 < 8 and 2x - 6 > -8 requires algebraic operations to find the range of x. Once simplified, these inequalities yield clear cut boundaries for x (x < 7 and x > 1, respectively) and these form the solution set for the variable.

Piecewise expressions are functions defined by different expressions over different intervals. Absolute value functions often lead to piecewise definitions because they behave differently depending on the input value's relation to zero.

Inequality solutions, especially those involving absolute values, can be thought of in a piecewise manner because you are effectively considering different conditions (or 'pieces' of the domain), such as those before and after crossing 0. In the exercise, 2x - 6 < 8 and 2x - 6 > -8 represent two pieces. These lead to their respective inequalities which are then combined to find where they overlap, or intersect, to identify a 'piece' of the number line where both conditions are true.