¿Cómo puedo simplificar este ejercicio?

Solución

Propongo lo siguiente :

<img src="https://tex.z-dn.net/?f=1-%5Cfrac%7B1%7D%7Bx%7D%3D%5Cfrac%7Bx-1%7D%7Bx%7D%5C%5C-1%2B%5Cfrac%7B3%7D%7B1-%5Cfrac%7B1%7D%7Bx%7D%7D%3D-1%2B%5Cfrac%7B3%7D%7B%5Cfrac%7Bx-1%7D%7Bx%7D%7D%3D%5Cfrac%7B-%5Cfrac%7Bx-1%7D%7Bx%7D%2B3%7D%7B%5Cfrac%7Bx-1%7D%7Bx%7D%7D%3D%5C%5C%5Cfrac%7B-%28x-1%29%2B3x%7D%7B%5Cfrac%7Bx%7D%7B%5Cfrac%7Bx-1%7D%7Bx%7D%7D%7D%3D%5Cfrac%7B-x%2B1%2B3x%7D%7B%5Cfrac%7Bx%7D%7B%5Cfrac%7Bx-1%7D%7Bx%7D%7D%7D%3D%5C%5C%5Cfrac%7B2x%2B1%7D%7B%5Cfrac%7Bx%7D%7B%5Cfrac%7Bx-1%7D%7Bx%7D%7D%7D%3D%5Cfrac%7Bx%282x%2B1%29%7D%7Bx%28x-1%29%7D%3D%5Cfrac%7B2x%2B1%7D%7Bx-1%7D%5C%5C%5Ctextup%7BFinalmente%3A%7D%5C%5C%5Cfrac%7B1%7D%7B%5Cfrac%7B2x%2B1%7D%7Bx-1%7D%7D%3D%5Cfrac%7Bx-1%7D%7B2x%2B1%7D" />

Saludos.