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When simplifying expressions, particularly those involving absolute values, the aim is to reduce the expression to its simplest form. This involves carrying out any arithmetic operations inside the absolute value signs, such as addition or subtraction, before dealing with the absolute value itself. The absolute value symbol (| |) represents the distance of a number from zero on the number line, regardless of direction, so it always results in a non-negative number.

For example, when given the expression \( |3 - 14| \), we subtract 14 from 3 to get \( |-11| \), which is equivalent to \( | - (14 - 3) | \). Since the absolute value of -11 is 11 (the distance from 0 without considering direction), the expression simplifies to just 11. Always remember, irrespective of the signs inside the absolute value brackets, the outcome after removing the absolute value will be positive or zero.

A number line is a visual representation of numbers laid out in a straight line, where each point corresponds to a number. It usually starts with zero in the middle, with positive numbers to the right and negative numbers to the left. Using a number line can help with understanding the concept of absolute value, as it visually displays the distance of any number from zero.

Take the number -11 for instance; on the number line, it is 11 units left of zero. When we find the absolute value of -11, we're not changing its position on the number line but rather asking how far away it is from zero, ignoring the direction. Therefore, on the number line, the absolute value is represented by the positive distance, making \( |-11| = 11 \). This understanding aids tremendously in visualizing why absolute value always results in a non-negative value.

Positive numbers are the set of numbers greater than zero. They can be found to the right side of zero on a number line. These numbers are important in the context of absolute values because the absolute value of any number, whether it's positive, negative, or zero, will always be a positive number or zero itself.

Being positive is the inherent nature of absolute values. For instance, the absolute value of -11 mirrors the positive number 11 since we're looking at the distance it represents on the number line without considering its negative sign. This is a fundamental property of absolute value and reflects the fact that distance itself cannot be negative; likewise, absolute values cannot be negative either.