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Try removing the batteries, and leaving the camera on a (dry, non-humid, non-dusty) shelf for overnight. Then put the batteries back in and try the camera again.
Then try cleaning the terminals in the battery compartment, after removing the batteries, first by rubbing off with a fresh pencil eraser, then by wiping the terminals off with cotton swabs moistened in rubbing alcohol, then letting them dry off for 5+ minutes with the battery compartment door open, then try turning the camera On again.
How long has the battery or batteries in the camera been in-use? If over a year, then you might possibly have a battery/batteries that are getting weak. Rechargeable batteries do get weaker with time, and eventually fail to deliver enough power to start up a device like a digital camera; they then need replacing. The replacement battery needs to be charged up for 6+ hours before its first use, owing to its having been on a shelf for a considerable time.
How old is the camera? in months or years since you bought it.
Has the camera been dropped or otherwise subjected to mechanical shock(s) in its lifetime?
Has the camera been operated in or subjected to, temperatures below 32 degrees F (freezing point) or above about 100 degrees F ?
Has it ever been subjected to a humidity higher than 85 % or gotten wet ?
Has the camera been operated in a gritty or sandy environment, such as a beach, a seaside, a construction site, or the like?
Alternating batteries does eliminate the problem of having to cease operations owing to having no charged battery left... but, even alternating batteries do age.
How old are the set of alternating batteries ? did they get bought (or come with) the camera, in 2009 ?
The batteries are old enough to have failed to deliver enough power, or to produce enough voltage. I recommend trying a replacement battery:
Rechargeable batteries do get weaker with time, and eventually fail to deliver enough power to start up a device like a digital camera; they then need replacing.
The replacement battery needs to be charged up for 6+ hours before its first use, owing to its having been on a shelf for a considerable time.
You can let the case rest for a while, until there is further news.
But if this closes the case as far as you are concerned, please rate my answer, so that the case may be closed. Thanks.
OK. Monday or Tuesday is fine. Have a good trip.