We give a new inequality between the energy of a graph and a weighted sum over the edges of the graph. Using this inequality we prove that E ( G ) ≥ 2 R ( H ) , where E ( G ) is the energy of a graph G and R ( H ) is the Randić index of any subgraph of G (not necessarily induced). In particular, this generalizes well-known inequalities E ( G ) ≥ 2 R ( G ) and E ( G ) ≥ 2 μ ( G ) where μ ( G ) is the matching number. We give other inequalities as applications to this result.