The first sentence, 'Jack felt nauseated because he had eaten a hamburger at a restaurant that was greasy,' is the best choice. It clearly indicates that the restaurant is greasy, helping to explain Jack's nausea. The other sentences lack the same level of clarity through their modifiers.
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The first sentence clearly uses modifiers to indicate that the restaurant was greasy, providing clarity on Jack's feeling nauseated. The other sentences do not provide the same level of descriptive clarity. Therefore, the first option is the best choice. ;