Machine unlearning aims to efficiently eliminate the influence of specific training data, known as the forget set, from the model. However, existing unlearning methods for Large Language Models (LLMs) face a critical challenge: they rely solely on negative feedback to suppress responses related to the forget set, which often results in nonsensical or inconsistent outputs, diminishing model utility and posing potential privacy risks. To address this limitation, we propose a novel approach called Alternate Preference Optimization (AltPO), which combines negative feedback with in-domain positive feedback on the forget set. Additionally, we introduce new evaluation metrics to assess the quality of responses related to the forget set. Extensive experiments show that our approach not only enables effective unlearning but also avoids undesirable model behaviors while maintaining overall model performance. Our implementation can be found at https://github.com/molereddy/Alternate-Preference-Optimization.