Can I safely eat whatever goes through the x-ray machine?

There is

• radiation that can only heat,
• radiation that can additionally ionize,
• and radiation that can actually make stuff radioactive.

XRays are the second kind, ionizing radiation, meaning they may alter some molecules (i.e. the arrangement of atoms), but will not affect the atoms themselves(so no radioactivity is created).

The altering of molecules may also happen to the DNA molecule, which is why XRays are kept to a minimum. So the apple sent through the machine might get a mutation, but the chance of that being a problem to the apple are remote, and the mutated apple being a problem to you is virtually nonexistent. Sending old school films through an old school XRay might be a problem for the film (film is coated with molecules that are easily altered, because detecting (visible)radiation is it's job, and old XRays used higher power sources).

Your association of XRay machines with radioactivity might come from the hazard sign above, that is sometimes quite prominently displayed on Xray machines.

It simply warns of ionizing radiation, which can, as stated, damage your DNA, thereby giving you cancer, etc, depending on strength. It's popular meaning of "Danger! Radioactivity" came about because radioactive materials emit ionizing radiation (that's actually why it's called radioactive, the ionizing radiation messes with radio-equipment). So Radioactive materials emit ionizing radiation, but ionizing radiation does not produce radioactivity.

Yes, you can safely eat food that's been through the X-ray machine (assuming, of course, that it was safe to eat before going through the nuker).

There are two main ways in which something not previously radioactive (such as the food in your luggage) can be made radioactive by electromagnetic radiation (such as the X-rays probing your luggage): photodisintegration and photofission. Photodisintegration requires photons¹ with energies in the MeV (megaelectronvolt)² range, while X-ray luggage scanners use radiation "in the low-to-medium keV [kiloelectronvolt] energy range", a couple orders of magnitude feebler. Therefore, photodisintegration is not a concern here.³ As for photofission, it only occurs to any detectable degree for things that're already prone to fissioning, so, unless you're going to be eating something like plutonium or uranium-235, photofission shouldn't affect your food either.

Conclusion: your food will not be any more radioactive when it comes out of the X-ray machine than it was when it went in.

¹: Photons are the basic units of electromagnetic radiation, such as visible light, X-rays, radio rays, ultraviolent rays, infrared rays, etc., etc., etc..

²: An electronvolt (eV) is a measure of energy; it is defined as the amount by which the energy of an electron changes when it moves through an electrical potential difference of one volt. A kiloelectronvolt (keV) equals one thousand electronvolts; a megaelectronvolt (MeV) equals one million electronvolts. The more (kilo-/mega-)electronvolts an X-ray photon has, the more energy it carries, and the more damage it can do with that energy.

³: Some specialised scanners for things like large cargo containers do use MeV-range X-rays, but that isn't a concern for the food in your carry-on luggage.

You might be surprised to know this, but you'll get far more radiation exposure on the airplane than your food gets exposed to in the x-ray machine

As the diagram makes clear, walking through an airport security scanner exposes a person to about the same ionizing radiation dose as eating a banana. Flying from New York to Los Angeles exposes you to roughly the same amount of radiation you'd get from eight dental X-rays — and less than you'd get living in a stone house for a year. And those peanuts that airlines hand out? They're a little radioactive, too.

"Radiation is one example of where people have such a wrong idea about what is dangerous, and are also unaware of its ubiquitous nature," says Barish. "Radiation is all around us. It is in us."

There's this handy chart(referenced above) which shows that neither is particularly dangerous on a dosage scale. If you regularly fly on flights passing over the poles, you might need to be more concerned (but even then we're talking LOTS of flights to get a concerning dose).

So eat up. There's no danger from the x-ray machine.