Quantum relative entropy appears in many forms throughout quantum information. Though asymmetric in its arguments, quantum relative entropy is often used analogously to a distance between densities. Bounds on linear combinations of quantum relative entropies between different densities are valuable but challenging to prove even in finite dimension. Here I present several such inequalities, including a triangle-like inequality and a multiplicative, perturbation-like bound in terms of the relative entropy's second argument. These yield a tightened form of inequality known as quasi-factorization or approximate tensorization, which in turn has applications to decay rates of quantum Markov semigroups. I highlight the use of quasi-factorization in entropic uncertainties for finite-dimensional systems and in estimating decay rates for Markov processes described by finite graphs.