Although improvements have been made in the performance of sentiment analysis tools, the automatic detection of negated text (which affects negative sentiment prediction) still presents challenges. More research is needed on new forms of negation beyond prototypical negation cues such as "not" or "never." The present research reports findings on the role of a set of words called "approximate negators," namely "barely," "hardly," "rarely," "scarcely," and "seldom," which, in specific occasions (such as attached to a word from the non-affirmative adverb "any" family), can operationalize negation styles not yet explored. Using a corpus of 6,500 tweets, human annotation allowed for the identification of 17 recurrent usages of these words as negatives (such as "very seldom") which, along with findings from the literature, helped engineer specific features that guided a machine learning classifier in predicting negated tweets. The machine learning experiments also modeled negation scope (i.e. in which specific words are negated in the text) by employing lexical and dependency graph information. Promising results included F1 values for negation detection ranging from 0.71 to 0.89 and scope detection from 0.79 to 0.88. Future work will be directed to the application of these findings in automatic sentiment classification, further exploration of patterns in data (such as part-of-speech recurrences for these new types of negation), and the investigation of sarcasm, formal language, and exaggeration as themes that emerged from observations during corpus annotation.