Inspiron 15-5559 Wont boot, battery light blinks 2 amber then 4 white, then repeats

You can try one module at a time in each socket -- if one module fails in either socket, but the other is OK in both, replace that module.  If both modules fail in one socket but not the other, the socket is bad.  You can use the system with a single module, but replacing the system board is necessary to solve the problem.  See below.

If neither module works in either socket, the chances of two module failures is low relative to that of the mainboard being the problem.  How much do you want to spend on repairing a system that is worth perhaps $50-75 complete and working?  You're likely better off replacing rather than repairing the system.

Thanks ejn63.

I tried one memory module at a time in each socket.  The sockets are labeled A and B, so lets call the memory modules 1 and 2.  The 1A and 1B combinations both worked fine, so it looked like a bad memory module 2 at first.  So I tried 2A, and it worked fine.  2B failed.  So maybe it is just one memory module incompatible with one socket?  

So I tried 2A and 1B together, failed.  So I removed 2A and just 1B failed.  So I went back to just 1A, and it failed.  

Now I will confess that when I started testing, I was removing the battery and pushing the power button for 30 seconds between each test, but to save time, I stopped those steps for a bit.  When I went back to removing the battery and pushing the power button for 30 seconds between each test I got pretty consistent results of either module working fine in the A Socket, but getting the 2 amber and 4 white flashing lights pretty consistently with either module in Socket B.

One observation, to remove the battery, push the power button, reinsert the battery, switch memory modules around, then power on the laptop involves closing the laptop, flipping the laptop over to close the battery, then flipping it back and opening it to push the power button, then closing and flipping it over again to switch around memory modules and replace battery, then flipping it back and reopening to try turning it on again.  That is a lot of acrobatics for a 9 year old laptop, so if there is a loose or flaky connection somewhere on the motherboard, or even something loose that is intermittently shorting/opening a circuit, all of that movement and possibly flexing of the motherboard could be the source of the intermittent results.

I do have backups of the laptop, but I am going to copy all of the data off of it ASAP, and limit it's use to relatively static conditions, such as on a dining room table with all data immediately going onto a Server or the Cloud.  No more carrying it around a worksite(which I had only done a couple of times recently).

Thanks for your troubleshooting help!