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In mathematics, non-negative numbers are numbers that are either positive or zero. In simpler terms, they do not have a negative sign in front of them.
For instance, 0, 1, 2, and 3 are all non-negative numbers. They can be written as 0, +1, +2, and +3, respectively.
Think of non-negative numbers as being on or to the right of zero on the number line.
The term 'non-negative' is essential when discussing absolute values because it helps to understand why the absolute value of a number can never be negative. Absolute value measures how far a number is from zero, without considering direction. Thus, it makes sense that this distance can never be negative.

The concept of absolute value is closely tied to the idea of distance on the number line.
Picture a number line, with zero at its center. The distance between two points on this line is always a non-negative value.
For example, if you measured the distance from 3 to zero, the answer would be 3 units. Likewise, the distance from -4 to zero is 4 units.
It doesn't matter whether the number is positive or negative; the distance itself (absolute value) is always positive or zero.
When we say \( |x| \), we are talking about the distance between the point corresponding to 'x' and the origin (zero) on the number line.
This is why \( |x| \) is never negative, as distance cannot be negative.

In algebra, the absolute value of a number is defined using the following notation: \( |x| \).
This is read as 'the absolute value of x'.
Mathematically, the absolute value of x is defined as: \[ |x| = \begin{cases} x & \text{if } x \text greater than or equal to 0 \ -x & \text{if } x \text less than 0 \end{cases} \]
This definition means, if x is a positive number or zero, its absolute value is simply 'x'.
However, if x is a negative number, its absolute value is 'x' multiplied by -1, making it a positive value.
These algebraic rules ensure that the absolute value, or the distance, is always a non-negative number, reinforcing why \( |x| \) can never be negative.