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For all real numbers x and y, |x-y| is e...

For all real numbers x and `y, |x-y|` is equivalent to which of the following?

A

`x+y`

B

`sqrt(x-y)`

C

`(x-y)^(2)`

D

`sqrt((x-y)^(2))`

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem of determining which expression is equivalent to |x - y| for all real numbers x and y, we can follow these steps: ### Step-by-Step Solution: 1. **Understanding the Absolute Value**: The expression |x - y| represents the absolute value of the difference between x and y. This means it measures the distance between x and y on the number line, regardless of which one is larger. 2. **Setting Up the Equation**: Let’s denote |x - y| as z. Thus, we have: \[ z = |x - y| \] 3. **Squaring Both Sides**: To eliminate the absolute value, we can square both sides of the equation: \[ z^2 = (|x - y|)^2 \] 4. **Using the Property of Absolute Values**: The property of absolute values states that: \[ |a|^2 = a^2 \] Therefore, we can rewrite the equation as: \[ z^2 = (x - y)^2 \] 5. **Taking the Square Root**: Since z is defined as |x - y|, we can express z in terms of x and y: \[ z = \sqrt{(x - y)^2} \] 6. **Final Conclusion**: Thus, we can conclude that: \[ |x - y| = \sqrt{(x - y)^2} \] This matches with option (d) from the given choices. ### Final Answer: The expression |x - y| is equivalent to: \[ \text{Option (d): } \sqrt{(x - y)^2} \]
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