ar \( \frac{(x-y)^{2}}{2} \div \frac{2}{3(y-x)^{-2}}=3 / 4 \)

Let’s simplify the expression step by step! First, rewrite the equation: \[ \frac{(x-y)^{2}}{2} \div \frac{2}{3(y-x)^{-2}} = \frac{3}{4} \] Now, dividing by a fraction is the same as multiplying by its reciprocal. Hence, \[ \frac{(x-y)^{2}}{2} \times \frac{3(y-x)^{2}}{2} = \frac{3}{4} \] Notice that \(y - x = -(x - y)\). Therefore, \((y - x)^{2} = (x - y)^{2}\). So, let's plug that back into the equation: \[ \frac{(x-y)^{2}}{2} \times \frac{3(x - y)^{2}}{2} = \frac{3}{4} \] Now, we can combine the terms: \[ \frac{3(x - y)^{4}}{4} = \frac{3}{4} \] Next, multiply both sides by 4 to eliminate the denominator: \[ 3(x - y)^{4} = 3 \] Now, divide both sides by 3: \[ (x - y)^{4} = 1 \] Taking the fourth root of both sides: \[ x - y = \pm 1 \] Thus, the solutions are \(x - y = 1\) or \(x - y = -1\) which translates to: 1. \(x = y + 1\) 2. \(x = y - 1\) And that's the solution to the given equation!