If $a - 8 = b$, then determine the value of $|a - b| - |b - a|$.

### Concept & Formula This problem tests the fundamental property of absolute values related to distance. The absolute value $|x - y|$ represents the distance between point $x$ and point $y$ on a number line. Because distance is scalar and directionless, the distance from $x$ to $y$ is exactly the same as the distance from $y$ to $x$. $$ \text{Absolute Value Identity: } |x - y| = |y - x| \text{ for any real numbers } x, y $$ ### Step-by-Step Solution * **Given:** * Equation: $a - 8 = b$ * Target expression: $|a - b| - |b - a|$ * **Calculation:** 1. Rearrange the given equation to find the value of $a - b$. $a - 8 = b \implies a - b = 8$. 2. Rearrange the equation to find the value of $b - a$. Multiply the equation $a - b = 8$ by $-1$. $-1(a - b) = -1(8) \implies b - a = -8$. 3. Substitute these values into the target expression: $|a - b| - |b - a|$. $|8| - |-8|$. 4. Evaluate the absolute values. The absolute value of a number is its non-negative distance from zero. $|8| = 8$ $|-8| = 8$ 5. Subtract the results: $8 - 8 = 0$. ### Exam Strategy & Shortcut You do not need to calculate the actual values of $a - b$ or $b - a$. By definition, $|X| = |-X|$. Since $(a - b)$ is simply the negative of $(b - a)$, their absolute values will always be identical regardless of what numbers $a$ and $b$ are. Subtracting two identical values always yields $0$. You can ignore the equation $a - 8 = b$ entirely; it is just a distractor. ### Common Pitfall Students often get confused by the negative signs inside the absolute value brackets and mistakenly calculate $|8| - (-8) = 16$. Remember that absolute value operations must be fully resolved into positive numbers *before* applying the subtraction sign that sits between them. ### Final Answer Therefore, the correct answer is 0.