Mojave

ATI Radeon HD 5770 1024 MB compatibility with Mojave?

The 5770 does not support Metal.

macOS 10.14 Mojave - Technical Specifications:

https://support.apple.com/kb/SP777

Upgrade to macOS Mojave - Official Apple Support

https://support.apple.com/macos/mojave

https://itunes.apple.com/us/app/macos-mojave/id1398502828?mt=12

MacBook (Early 2015 or newer)

MacBook Air (Mid 2012 or newer)

MacBook Pro (Mid 2012 or newer)

Mac mini (Late 2012 or newer)

iMac (Late 2012 or newer)

iMac Pro (2017)

Mac Pro (Late 2013; Mid 2010 and Mid 2012 models with recommended Metal-capable graphics cards)

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macOS 10.13 High Sierra - Technical Specifications:

https://support.apple.com/kb/SP765?locale=en_US

How to upgrade to macOS High Sierra

https://support.apple.com/en-us/HT208969

MacBook (Late 2009 or newer)

MacBook Pro (Mid 2010 or newer)

MacBook Air (Late 2010 or newer)

Mac mini (Mid 2010 or newer)

iMac (Late 2009 or newer)

Mac Pro (Mid 2010 or newer)