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In this paper, we refine our method for the abundance analysis of high-resolution spectroscopy of the integrated light of unresolved globular clusters (GCs). This method was previously demonstrated for the analysis of old (>10 Gyr) Milky Way (MW) GCs. Here, we extend the technique to young clusters using a training set of nine GCs in the Large Magellanic Cloud. Depending on the signal-to-noise ratio of the data, we use 20-100 Fe lines per cluster to successfully constrain the ages of old clusters to within a ~5 Gyr range, the ages of ~2 Gyr clusters to a 1-2 Gyr range, and the ages of the youngest clusters (0.05-1 Gyr) to a ~200 Myr range. We also demonstrate that we can measure [Fe/H] in clusters with any age less than 12 Gyr with similar or only slightly larger uncertainties (0.1-0.25 dex) than those obtained for old MW GCs (0.1 dex); the slightly larger uncertainties are due to the rapid evolution in stellar populations at these ages. In this paper, we present only Fe abundances and ages. In the next paper in this series, we present our complete analysis of ~20 elements for which we are able to measure abundances. For several of the clusters in this sample, there are no high-resolution abundances in the literature from individual member stars; our results are the first detailed chemical abundances available. The spectra used in this paper were obtained at Las Campanas with the echelle on the du Pont Telescope and with the MIKE spectrograph on the Magellan Clay Telescope. [ABSTRACT FROM AUTHOR]