To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-3\right)\left(x-5\right)$ and $\left(x-3\right)\left(x-1\right)$ is $\left(x-5\right)\left(x-3\right)\left(x-1\right)$. Multiply $\frac{1}{\left(x-3\right)\left(x-5\right)}$ times $\frac{x-1}{x-1}$. Multiply $\frac{1}{\left(x-3\right)\left(x-1\right)}$ times $\frac{x-5}{x-5}$.
Since $\frac{x-1}{\left(x-5\right)\left(x-3\right)\left(x-1\right)}$ and $\frac{x-5}{\left(x-5\right)\left(x-3\right)\left(x-1\right)}$ have the same denominator, add them by adding their numerators.
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-3\right)\left(x-5\right)$ and $\left(x-3\right)\left(x-1\right)$ is $\left(x-5\right)\left(x-3\right)\left(x-1\right)$. Multiply $\frac{1}{\left(x-3\right)\left(x-5\right)}$ times $\frac{x-1}{x-1}$. Multiply $\frac{1}{\left(x-3\right)\left(x-1\right)}$ times $\frac{x-5}{x-5}$.
Since $\frac{x-1}{\left(x-5\right)\left(x-3\right)\left(x-1\right)}$ and $\frac{x-5}{\left(x-5\right)\left(x-3\right)\left(x-1\right)}$ have the same denominator, add them by adding their numerators.
