Each Route 53 hosted zone is going to have it's own name servers. So for each of example.com and example-x.com, you need to set the name server settings at your registrar to be the name servers displayed on each hosted zone.
set the name servers for example-x.com to be that of example.com
Most likely, the name servers will be different, so this is the wrong thing to do. Treat the name server settings for each domain independently.
Once the name servers are setup correct at your registrar, then the next thing is to create an A record in your example-x.com hosted zone for example-x.com. The value of this A record should be your Elastic IP address.
Once this is set, if you
ping example-x.com, it should resolve to your elastic IP. It's irrelevant at this time if the ping actually succeeds or not. What we care about is whether we get the IP address.
At this point, it's up to your nginx and web server configuration to take over whether the redirects happen correctly.
One alternative option is to use Amazon S3 to handle your domain redirects. For example, you can create an S3 bucket named example.com. In the bucket configuration you would have it redirect to www.example.com. In your Route 53 hosted zone for example.com, you would make an ALIAS record for example.com to point to your S3 bucket.
Doing the above means you don't have to configure any of the redirects within your nginx or web server configuration: everything that reaches your actual web server is already using www.example.com.