The entire problem comes from when the headphones are unplugged and the computer doesn't have a way to output the sound. This causes the "headphone device" to "break" and when you plug headphones back in, it needs to troubleshoot and recognize the device again.
First ensure that the volume is not muted. Do so by traveling to the volume controls near the clock on your task bar or checking the volume in whichever program you are using.
If your microphone icon has a red x over it, then the port is "broken" and needs to be fixed. No worries, though! Microsoft can do this on its own. Simply click the icon and Microsoft will automatically begin troubleshooting the problem. It is easiest if you plug your headphones back into the jack during this troubleshooting time, so that the jack can recognize a suitable output and repair itself.
If the problem persists, try rebooting your computer with the headphones plugged in. This is a last resort, but it should fix the problem.
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