Complex fraction made of sums involving rational expressions: Problem... Simplify. \( \frac{\frac{5}{v-2}-\frac{1}{v+6}}{\frac{6}{v+6}-\frac{2}{v-2}} \)

First, let's tackle the numerator: \( \frac{5}{v-2} - \frac{1}{v+6} \). To combine these fractions, we find a common denominator, which is \((v-2)(v+6)\). This gives us: \[ \frac{5(v+6) - 1(v-2)}{(v-2)(v+6)} = \frac{5v + 30 - v + 2}{(v-2)(v+6)} = \frac{4v + 32}{(v-2)(v+6)} \] Now, for the denominator: \( \frac{6}{v+6} - \frac{2}{v-2} \). The common denominator here is also \((v-2)(v+6)\). Therefore, we have: \[ \frac{6(v-2) - 2(v+6)}{(v-2)(v+6)} = \frac{6v - 12 - 2v - 12}{(v-2)(v+6)} = \frac{4v - 24}{(v-2)(v+6)} \] Now, substituting the results back into the complex fraction: \[ \frac{\frac{4v + 32}{(v-2)(v+6)}}{\frac{4v - 24}{(v-2)(v+6)}} \] This simplifies to: \[ \frac{4v + 32}{4v - 24} \] Now, we can factor both the numerator and the denominator. The numerator factors to \(4(v + 8)\) and the denominator factors to \(4(v - 6)\). Therefore, cancelling the common factor of 4, we get: \[ \frac{v + 8}{v - 6} \] Thus, the simplified form of the complex fraction is: \[ \frac{v + 8}{v - 6} \]