If you have a Drupal installation in a subdirectory and change it to another subdirectory the image paths are going to be incorrect, the path sees the initial subdirectory as the root folder.
You can get around this in 2 ways, one is to use .htaccess file to redirect the old directory to the new one but a preferred method would be to hardcode the new path back in to all the images. Rather than a painstaking troll though all the content a find and replace in the mysql is your best friend.
This can be achieved in phpMyAdmin.
Select the Drupal Database and have a look at the 'field_data_body' table, and the 'body_value' field if you have used your text editor to insert images you should see the source 'src' attribute in the html markup.
So now you can add some SQL to replace that part of the string with the new path, click in the SQL tab add the SQL and click Go, SQL to add is:
UPDATE field_data_body SET body_value = REPLACE(body_value,'src="/olddirectory','src="/newdirectory');
Flush the Drupal cache, that's it all the images will be updated and have the new path.