"Help me with my old MacBook air: I’m trying to start up my old MacBook Air but it lets me sign in then it just sits there running and looks like this constantly running with this symbol on the screen I’ve tried to do command-shift-R nothing I’ve tried all the things mentioned so please help"
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Get macOS Reinstalled:
Have Ethernet?
If so, then try connecting and performing it through that. With no Ethernet Port, and only USB-A or -C ports, you'll need one of the following adapters. Being an Earlier Mac model, you'll likely need a USB-A 2.0 to Ethernet Adapter (being obsolete, it is no longer available on Apple's Web site. You'll have to get one from a third-party, perhaps ebay, Amazon, or craigslist.).
A. For More Modern Macs (2014 and after):
B. For Earlier Mac Models: