Q: upgrade from 10.5.8

Go to Apple Menu (Top left) About this Mac > More Info > System Report and copy and paste the Hardware Overview back here to see if the iMac is compatable first.

Thanks

Pete

To upgrade beyond 10.5.8, you need to purchase the Snow Leopard install DVD

After you upgrade to 10.6.3, you update to 10.6.8

10.9 is Mavericks

Allan

Read the hardware requirements.

   Snow Leopard Purchase    

Check that your computer is compatible with Mountain Lion/Mavericks.

To check the model number hold down the option/alt key, go to the Apple menu and select System Information.

• iMac (Mid 2007 or newer) model number 7,1 or higher

Check to make sure your applications are compatible. PowerPC applications are no longer supported after 10.6.      

Application Compatibility

Applications Compatibility (2)

Eric Root wrote:

 

 

To check the model number hold down the option/alt key, go to the Apple menu and select System Information.

 

Neat trick! Haven't seen that one before. Much easier. Thanks

Pete

Your iMac is from mid-2008 and therefore could potentially run Mavericks.  See:

https://www.everymac.com/ultimate-mac-lookup/?search_keywords=iMac8,1

http://support.apple.com/kb/HT5842

Buy the Snow Leopard OS (link in post above). Before doing anything to upgrade make a backup. After booting from the Snow Leopard DVD, I would run Disk Utility Repair Permissions and Repair Disk. Then as Allan posted, install Snow Leopard  and update it to 10.6.8. Make sure the computer is working properly, then do another backup. Go to the App Store, download Mavericks 10.9.2, and install it. Note - the installer will self destruct after the installation, so if you want to keep a spare copy, duplicate it and move it out of the Applications folder. Run Software Update to make sure everything is up-to-date.

You are welcome.

Correct.