To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-3\right)\left(x-4\right)$ and $\left(x-4\right)\left(x-5\right)$ is $\left(x-5\right)\left(x-4\right)\left(x-3\right)$. Multiply $\frac{1}{\left(x-3\right)\left(x-4\right)}$ times $\frac{x-5}{x-5}$. Multiply $\frac{1}{\left(x-4\right)\left(x-5\right)}$ times $\frac{x-3}{x-3}$.
Since $\frac{x-5}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ and $\frac{x-3}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ have the same denominator, add them by adding their numerators.
Since $\frac{2}{\left(x-5\right)\left(x-3\right)}$ and $\frac{1}{\left(x-5\right)\left(x-3\right)}$ have the same denominator, add them by adding their numerators. Add $2$ and $1$ to get $3$.
To add or subtract expressions, expand them to make their denominators the same. Least common multiple of $\left(x-3\right)\left(x-4\right)$ and $\left(x-4\right)\left(x-5\right)$ is $\left(x-5\right)\left(x-4\right)\left(x-3\right)$. Multiply $\frac{1}{\left(x-3\right)\left(x-4\right)}$ times $\frac{x-5}{x-5}$. Multiply $\frac{1}{\left(x-4\right)\left(x-5\right)}$ times $\frac{x-3}{x-3}$.
Since $\frac{x-5}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ and $\frac{x-3}{\left(x-5\right)\left(x-4\right)\left(x-3\right)}$ have the same denominator, add them by adding their numerators.
Since $\frac{2}{\left(x-5\right)\left(x-3\right)}$ and $\frac{1}{\left(x-5\right)\left(x-3\right)}$ have the same denominator, add them by adding their numerators. Add $2$ and $1$ to get $3$.
