Social media platforms like X (formerly Twitter) generate vast volumes of user-generated content that provide real-time insights into public sentiment. Despite the widespread use of traditional machine learning methods, their limitations in capturing contextual nuances in noisy social media text remain a challenge. This study leverages the Sentiment140 dataset, comprising 1.6 million labeled tweets, and develops predictive models for binary sentiment classification using Naive Bayes, Logistic Regression, and the transformer-based BERT model. Experiments were conducted on a balanced subset of 12,000 tweets after comprehensive NLP preprocessing. Evaluation using accuracy, F1-score, and confusion matrices revealed that BERT significantly outperforms traditional models, achieving an accuracy of 89.5% and an F1-score of 0.89 by effectively modeling contextual and semantic nuances. In contrast, Naive Bayes and Logistic Regression demonstrated reasonable but consistently lower performance. To support practical deployment, we introduce SentiFeel, an interactive tool enabling real-time sentiment analysis. While resource constraints limited the dataset size and training epochs, future work will explore full corpus utilization and the inclusion of neutral sentiment classes. These findings underscore the potential of transformer models for enhanced public opinion monitoring, marketing analytics, and policy forecasting.