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Half reactions separate the oxidation and the reduction reactions into two half reactions. You can use the individual two half reactions to find out which elements are oxidized and reduced, and also balance redox reactions (since the number of electrons transferred should end up cancelling out.
By half reactions, Lavelle simply means to separate the oxidation reaction and the reduction reaction to balance one half reaction first, keep track of electrons exchanged and piece it back together with the other half reaction to fully balance the reaction.
Half reactions don't necessarily mean the reaction is split in half exactly. Oxidation and reduction reactions are two halves of a redox reaction. So, when you're breaking the reaction up, you're breaking the redox reaction into one oxidation and one reduction reaction.
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