Q: Can I upgrade my iMac 10.4.11 to 10.5 or 7?

Impossible to answer with information provided.

Allan

You will need to give us more info if you want a answer.

At least to Snow Leopard if your iMac only has a Core Duo processor. If you have a Core 2 Duo or better processor then you can upgrade all the way to Lion if you wish provided you meet the system requirements.

Snow Leopard General requirements

• Mac computer with an Intel processor
• 1GB of memory
• 5GB of available disk space
• DVD drive for installation
• Some features require a compatible Internet service provider; fees may apply.
• Some features require Apple’s MobileMe service; fees and terms apply.

Lion System Requirements

• Mac computer with an Intel Core 2 Duo, Core i3, Core i5, Core i7, or Xeon processor
• 2GB of memory
• OS X v10.6.6 or later (v10.6.8 recommended)
• 7GB of available space
• Some features require an Apple ID; terms apply.

You can purchase the Snow Leopard Retail DVD at a local Apple retailer or from the Online Store for $29.00. The Lion purchase from the App Store is $30.00. Lion is downloaded from the App Store so you need a decent internet connection. Or you can purchase a Lion USB flash drive installer from Apple for $69.00 and just skip buying Snow Leopard. If you decide to jump directly to Lion using the flash drive then be sure to back up your current system first. You will need to erase the drive before installing Lion.

How to Install OS X Updates Successfully

A. Repair the Hard Drive and Permissions:

Boot from your current OS X Installer disc. After the installer loads select your language and click on the Continue button. Then select Disk Utility from the Utilities menu. After DU loads select your hard drive entry (mfgr.'s ID and drive size) from the the left side list.  In the DU status area you will see an entry for the S.M.A.R.T. status of the hard drive.  If it does not say "Verified" then the hard drive is failing or failed. (SMART status is not reported on external Firewire or USB drives.) If the drive is "Verified" then select your OS X volume from the list on the left (sub-entry below the drive entry,) click on the First Aid tab, then click on the Repair Disk button. If DU reports any errors that have been fixed, then re-run Repair Disk until no errors are reported. If no errors are reported click on the Repair Permissions button. Wait until the operation completes, then quit DU and return to the installer. Now restart normally. 

If DU reports errors it cannot fix, then you will need Disk Warrior (4.0 for Tiger) and/or TechTool Pro (4.5.2 for Tiger) to repair the drive. If you don't have either of them or if neither of them can fix the drive, then you will need to reformat the drive and reinstall OS X.

B. Make a Bootable Backup Using Restore Option of Disk Utility:

1. Open Disk Utility from the Utilities folder.
2. Select the destination volume from the left side list.
3. Click on the Restore tab in the DU main window.
4. Check the box labeled Erase destination.
5. Select the destination volume from the left side list and drag it to the Destination entry field.
6. Select the source volume from the left side list and drag it to the Source entry field.
7. Double-check you got it right, then click on the Restore button.

Destination means the external backup drive. Source means the internal startup drive.

C. Important: Please read before installing:

1. If you have a FireWire hard drive connected, disconnect it before installing the update unless you will boot from this drive and install the update on it. Reconnect it and turn it back on after installation is complete and you've restarted.
2. You may experience unexpected results if you have installed third-party system software modifications, or if you have modified the operating system through other means. (This does not apply to normal application software installation.)
3. The installation process should not be interrupted. If a power outage or other interruption occurs during installation, use the standalone installer (see below) from Apple Downloads to update.  While the installation is in progress do not use the computer.

D. To upgrade:

Boot From The OS X Installer Disc:

1. Insert OS X Installer Disc into the optical drive.
2. Restart the computer.
3. Immediately after the chime press and hold down the "C" key.
4. Release the key when the spinning gear below the dark gray Apple logo appears.
5. Wait for installer to finish loading.

E. If updating:

1. Download and install update(s) 
2. Use Software Update, or
3. Download standalone updater(s).