Question

If \(x \neq 2y\), then \( \frac{x-2y}{2y-x} + \frac{2y-x}{x-2y} = \)

• A. 2(x-2y)
• B. 2y-x
• C. 1
• D. 0
• E. -2

Hint:

Whenever you see expressions of the form (a-b) and (b-a) in a fraction, remember that (b-a) = -(a-b). This means the fraction (a-b)/(b-a) will always simplify to -1 (as long as a ≠ b).

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
The problem involves simplifying an algebraic expression. The key is to recognize the relationship between the numerator and denominator in each fraction.

Step 2: Detailed Explanation:
Let's analyze the terms in the expression: \(x-2y\) and \(2y-x\).
Notice that one is the negative of the other. We can show this by factoring out -1.
\[ 2y-x = -(-2y+x) = -(x-2y) \] Now, let's substitute this into the given expression.
\[ \frac{x-2y}{2y-x} + \frac{2y-x}{x-2y} = \frac{x-2y}{-(x-2y)} + \frac{-(x-2y)}{x-2y} \] The condition \(x \neq 2y\) ensures that the denominators are not zero, so the fractions are well-defined.
Now, we can simplify each fraction.
\[ \frac{x-2y}{-(x-2y)} = -1 \] \[ \frac{-(x-2y)}{x-2y} = -1 \] So the expression becomes:
\[ -1 + (-1) = -2 \]

Step 3: Final Answer:
The value of the expression is -2.