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I'm running an AM3+ socket as well on my secondary setup, it has a Gigabyte GA-990FXA-UD3 and a FX-9590 overclocked to 5.2ghz. Most games seem to run fine on the 8350 and above (excluding unstable CPUs) but the 4000 and 6000 series of the FX line are not that great for gaming. My recommendation for you is getting a higher end 6000 series CPU, reasons below.
I'm going to go ahead and rule out the 4000 series CPUs first off due to the fact there is like a 67% difference between the 4000 series and the 8350 (the most stable and recommend for games FX CPU, going to compare the CPUs to this) in single core performance, so gaming on a 4000 series CPU is going to be extremely disappointingish (this is for current games such as Planetside 2 or Battlefield 4, for newer games, such as GTA V, it won't work at reasonable FPS at all). Stay away from these CPUs, they should be out of the question for gaming.
FX-6000 series CPUs are okayish for current games like Battlefield 4 on medium, give or take 40ish FPS with a decent video card, but they are not great. If you can go for a higher end 6000 series CPU, don't stop go for it. But I can't really give you an opinion on this series due to the fact that I completely skipped out on using this series. I have personally only used the 6300 and it is ok for current generation games but it will struggle with the newer games a lot, just because its lower end CPU.
In my opinion you should ditch both the 6000 series CPUs and go for the 8350, even if your current budget doesn't allow it. Just wait and save up until you can afford it and then go and buy it, it is well worth the extra price and the time you spent saving up for it. It's a great overclocker so it will also work for more newer generation games, but all in all it's up to you - good luck :).