Can you push content to both my personal and business Facebook pages?

We definitely can, but just so you know - if you invite your friends to like your business page, they will then see the articles twice in their news feed and might get annoyed (because they'll see the same articles coming from your personal page or your business page). This won't be a problem if you don't invite your friends to like the business page, but i would suggest inviting them and only pushing out to your business page. 

Inviting your friends to "like" the page is a great way to get initial likes on your page as well.