1. The problem is to understand the expression $|a-b|$ which represents the absolute value of the difference between $a$ and $b$. 2. The absolute value $|x|$ of a number $x$ is defined as: $$|x| = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases}$$ 3. Applying this to $|a-b|$, it means: $$|a-b| = \begin{cases} a-b & \text{if } a-b \geq 0 \\ -(a-b) & \text{if } a-b < 0 \end{cases}$$ 4. Simplifying the second case: $$-(a-b) = -a + b = b - a$$ 5. Therefore, $|a-b|$ is the distance between $a$ and $b$ on the number line, which is always non-negative. 6. In summary: $$|a-b| = \begin{cases} a-b & \text{if } a \geq b \\ b-a & \text{if } a < b \end{cases}$$ This ensures the result is always zero or positive, representing the magnitude of the difference between $a$ and $b$.