Run from a cloned startup drive
If you created a clone of your startup drive, you can boot from the clone and run Disk Utility from there. To do so, restart while holding down the Option key. From the screen that appears, select the cloned drive. When startup is complete, you'll find Disk Utility in the /Applications/Utilities folder, just as it is on your original drive.
You may be wondering: "Does my clone drive include a Recovery HD partition? Could I start up from that partition instead?" Maybe. If you used Shirt Pocket's $28 SuperDuper to make a clone, the clone will likely not have the Recovery HD partition. If you used Bombich Software's $40 Carbon Copy Cloner, it should. However, if you are using a cloned drive, I wouldn't bother with its Recovery HD partition in any case. Instead, boot from the drive directly, as I just described.
Try Safe Boot
To perform a Safe Boot, restart your Mac while holding down the Shift key. According to Apple, a Safe Boot "forces a directory check of the startup volume." This is essentially the same thing as running First Aid's Repair Disk. A downside of this method is that you get no feedback as to whether or not the repair succeeded. Still, if your problems vanish after doing a Safe Boot (and restarting again normally), you can assume that success was likely.
Access your Mac via Target Disk Mode
If you have two or more Macs, you may be able to connect one Mac to the other using Target Disk Mode. To do this, you'll need a cable that can connect the two Macs. For Macs with FireWire ports, that means an appropriate FireWire cable. For Macs with Thunderbolt ports, you'll want a Thunderbolt cable. If one Mac has FireWire and the other has Thunderbolt, you'll need a Thunderbolt to FireWire adapter.
Once connected, boot from the second (properly working) Mac and put the problem Mac in Target mode (by holding down the T key at startup). The Target Mac should now appear as an external drive to the startup Mac. You can now attempt to repair it via Disk Utility.
Boot from Internet Recovery Mode
Internet Recovery mode uses a combination of code stored in your Mac's firmware and a net-boot image stored on Apple's servers to boot your Mac.
To enter Internet Recovery mode, hold down the Command-Option-R keys at startup. Run Disk Utilityfrom there.
I would use this method only if you can't boot from the standard Recovery HD partition. This is because Internet Recovery mode requires that you download the needed software before it kicks into action. Depending on the speed of your Internet connection, this can take anywhere from about 5 minutes to more than 30 minutes. Also, note that Internet Recovery will not work with older Mac models.
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