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Networks provide useful tools for analyzing diverse complex systems from natural, social, and technological domains. Growing size and variety of data such as more nodes and links and associated weights, directions, and signs can provide accessory information. Link and weight abundance, on the other hand, results in denser networks with noisy, insignificant, or otherwise redundant data. Moreover, typical network analysis and visualization techniques presuppose sparsity and are not appropriate or scalable for dense and weighted networks. As a remedy, network backbone extraction methods aim to retain only the important links while preserving the useful and elucidative structure of the original networks for further analyses. Here, we provide the first methods for extracting signed network backbones from intrinsically dense unsigned unipartite weighted networks. Utilizing a null model based on statistical techniques, the proposed significance filter and vigor filter allow inferring edge signs. Empirical analysis on migration, voting, temporal interaction, and species similarity networks reveals that the proposed filters extract meaningful and sparse signed backbones while preserving the multiscale nature of the network. The resulting backbones exhibit characteristics typically associated with signed networks such as reciprocity, structural balance, and community structure. The developed tool is provided as a free, open-source software package.