Facebook Privacy – How to Hide Activities from Friend’s News Feed

Facebook keeps on performing some sort of A/B testing, where few users (based on some business logic of facebook) will see one UI, while rest will see other UI.
I don't know how it could be possible at the time this question was posted, but currently you can give a try to these steps:

1. On sidebar of the homepage, you will see 'triple dot' next to News Feed link.
2. Click on this. If you don't have this option, it means right now this feature is not available to you.
3. You will see multiple options here, like:
   i. Prioritize who to see first: Select names of friends whose posts you want to see on the top of your timeline.
   ii. Unfollow people to hide their post: You can unfollow your friends right from here, so that their post will never be displayed on your TL, unless you follow them again.
   iii. Reconnect with people you have unfollowed: If you had previously unfollowed someone, you can refollow them right from here.
   iv. Discover Pages that match your interest: Relevant pages based on your activity will be suggested by facebook. Click like to get their notification.
   v. See more options: If you have any hidden apps then you can manage them from here.
   
   Please have a look at the screenshot:
   

You could block the respective user(s).

If I share mutual friends with the user I am blocking, will they be able to see my comments to this mutual friend?

Blocking allows you to prevent most interactions with someone on Facebook. People you block won't be able to find you in searches, view your profile, or contact you with pokes, Wall posts, or personal messages. In addition, they will not be able to see your Wall posts, comments on mutual friends’ Walls, or that you are a member of the same group as them.

However, this might not work in certain situations:

Why am I able to see content added by a user that I've blocked on the site?

Blocking a user will prevent any type of direct interaction between you and that user. However, for technical reasons, there are certain instances when an individual you have blocked from your account may be able to view your comments on the site and vice versa, including comments on a mutual friend's photo. Please be assured that your listing will appear in black text, with no profile picture visible or contact links available.