It's important to me, but it's not a deal-breaker.

Like Tachikoma wrote, I prefer to play an older game on its original system, with the original controller. But it's still nice to have the option of backward compatibility. It's a nice gift to the consumers. It makes things less expensive for us, and less burdensome.

What you're saying basically becomes: I will never buy a Sony console because I can't play all the older games that I originally didn't show an interest in and have passed up on during their respective generation. Not hating, I just find it a bit silly to put the problem that way.

It's rather important for me.

I think its important during the first couple months or so of a new consoles life when top must play titles can be few and far between, and games are still releasing on the previous gen console

Not at all. If I want to play a PS2 game il turn my PS2 on.

ds has a huge library, kinda nice i can play them on my 3ds, that way i dont need to go out and buy ds just to play them.

so it works great for me

There are benefits to having it no doubt, but it shouldn't be the determining factor on whether you buy a console or not. Either you have a PS3 or you don't but want to have access to the back catalog. The former means you can just keep your PS3 handy and the later means you should have bought a PS3 and I don't know why it is taking you until the PS4 to do so.

Now as for a business decision it really comes down to what the difficulties are in doing so in which having the PS4 contain the hardware to run PS3 would require PS3 chips to be in the PS4 which would be expensive, think $100+ along with a bulkier, hotter, more energy demanding system, or require the PS4 to run on similar hardware as PS3, meaning the Cell architecture which would retain developmental difficulties and increase development costs further than the 10-15% increase we are looking at with x86 architecture which is amazing by the way.

For someone to argue that the PS4 is a no go because of the lack of backwards compatibility is ignoring all sense of reason. Having both a PS4 and a PS3 is not the end of the world and if you were looking to finance a PS4 by selling your PS3, chances are the increased costs of placing backward compatibility in the PS4 is going to eat up any financial advantage that would have given you.

Sometimes its the right thing to do, sometimes its not.

It is extremly important for me, I often buy and play games of older consol gens. Last Gen as I bought a Wii I bought 10 Gamecube Games as well for it, I bought Xbox 1 Games to play it on my Xbox 360 play Gameboy Advance on my Ds and now I often buy and play DS Games on my 3DS. So backwards compatibility is extremly important in my opinoin.

To me its essential because Sony is trying to entice me with both the PS3 and the PS4. If i compare the two, the PS3 is the better deal hands down. Now, if instead of a PS3 i could get the PS4 and get acess to my PS3 games, then it would be a more profitable solution for me and for Sony in the long term (as they would sell another PS4).

Since that isnt the case, and since new consoles launch with poor software offers until one year down the road, they usually have high defect rates and are incredibly expensive, i see no reason why i would go for a PS4 instead. The smart thing to do is to get a PS3 this year and a PS4 two years down the road. Now, you ponder wether this is on Sony's best interests or not.
Heck, i dont even mind if i have to buy an extra acessory for the backwards compatibility to work, aslong as i can have acess to the PS3 games aswell. On another note, isnt the PS4 so powerful that they could just emulate PS3 games on it? Its kind of odd to me.

If it is a dealbreaker that you can't play your PS3 games on PS4 then what are your other options? Buy an Xbox? Yeah those won't play PS3 either.