F. W. Gehring and B. G. Osgood [J. Anal. Math. 36, 50–74 (1979; Zbl 0449.30012)] proved that a \(K\)-quasiconformal map \(f\) between domains \(D\) and \(D'\) in \(\mathbb{R}^n\) satisfies a distortion inequality \[ \frac{d'(f(x),f(y))}{d'(f(x),\partial D')} \leq a_n \left(\frac{d(x,y)}{d(x, \partial D)}\right)^{\alpha} \] for all \(x \in D\) all \(y \in B(x, a_n^{-1/\alpha}d(x,\partial D)\) where \(d\) and \(d'\) are the quasihyperbolic metrics in \(D\) and \(D'\), respectively, \(a_n \in (0,1)\) depends only on \(n\) and \(\alpha = K^{1/(1-n)}\). The authors generalize this to metric measure spaces with \(Q\)-bounded geometries, which means that the Borel measures in these spaces satisfy \(r^{Q}/C_o \leq \mu(B(x,r)) \leq C_o r^{Q}\) and the \(Q\)-modulus of the path family joining to non-degenerate continua has a lower bound similar to the \(n\)-modulus in \(\mathbb{R}^n\) joining two such sets, see [J. Heinonen and P. Koskela, Acta Math. 181, No. 1, 1–61 (1998; Zbl 0915.30018)]. For a quasiconformal map \(f\) between proper domains \(D\) and \(D'\) in the metric spaces \(X\) and \(Y\) with \(Q\)-bounded geometries the distortion inequality takes the form \(d'(f(x),f(y)) \leq c \max((d(x,y)^{\alpha}, (d(x,y))\) for every \(x\) and \(y\) in \(D\) where \(c\) and \(\alpha\) depend on data. The proof makes use of the quasisymmetry property of a quasiconformal map between metric spaces with bounded geometries.