Conflict between package requirements when updating to Drupal 8.4
So, I've found a solution. I had to go into my composer.json file and specific these versions under require:

"drupal/core": "~8.4",
"drush/drush": "~9.0",


Once I updated those lines, I deleted the composer.lock file as well as the vendor directory. After that, I was able to install Drupal 8.4 using composer update --with-dependencies.