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Let's start by understanding the notion of absolute value. The absolute value of a number refers to the distance between that number and zero on a number line. This distance is always expressed as a non-negative number, as it represents a magnitude without regard to direction.

For example, evaluating the absolute value of -7, denoted as \( |-7| \), is simply figuring out how far -7 is from 0. This is akin to asking, 'How many steps would it take to get from -7 to 0 if each step covered one unit?' The answer is 7 steps, which means \( |-7| = 7 \).

The absolute value 'strips away' any sign the number might have, leaving only its size. If we take the absolute value of a positive number, it remains unchanged because it is already a distance from zero. For instance, \( |7| = 7 \), because 7 is already 7 units away from 0 on the number line. Whether the original number is negative or positive, its absolute value is always positive or zero.

The number line is an essential visual aid in mathematics that helps us understand the concepts of absolute value and inequalities better. It is a straight, horizontal line featuring evenly spaced marks representing integers, with zero at the center.

Using a number line, we can easily visualize the absolute value of any number. When we look for \( |-7| \), we simply count 7 units to the left of zero. Likewise, for \( |7| \) we count 7 units to the right. This spatial representation aids in grasping that both -7 and 7 are the same distance from zero, despite being on opposite sides.

Moreover, when we deal with inequalities, the number line lets us see the range of values that satisfy the inequality. By marking two points on the line and seeing which is closer to one end or the other, we can easily determine which inequality symbol to use. For the absolute values \( -|-7| \) and \( -|7| \) we clearly illustrate that they coincide on the same point, indicating equality. The number line, therefore, is not just a tool for solving problems but also for understanding the underlying concepts visually and intuitively.