Older LG Televison

The Apple TV 4K will work on any TV that has an HDMI input that is set for the same region that the Apple TV is set for. If your TV isn't 4K then the Apple TV will work with whatever the highest quality that your TV will support automatically. It will scan the device and the capabilities when it powers up and will select the highest quality automatically. If you don't want to use that quality that it selects, you can always go into the settings and change it to whatever quality you would like to use. It will show you only the quality that is compatible with your TV by default. If you select one that doesn't work, it will timeout if you don't select it on the next stage and go back to the one that was working so if you select one that doesn't work, it will go back to the one that was working

Would my LG 42LS3400-UA TV be compatible with Apple TV 4K?

Yes, it is compatible, but the resolution will be limited by the TV (1080p).