Choose a scheme $U$ and a surjective, flat, locally finite presented, and locally quasi-finite morphism $U \to \mathcal{X}$, see Theorem \ref{theorem-quasi-DM}. Set $R = U \times_\mathcal{X} U$. The projections $s, t : R \to U$ are surjective, flat, locally of finite presentation, and locally quasi-finite as base changes of the morphism $U \to \mathcal{X}$. There is a canonical morphism $[U/R] \to \mathcal{X}$ (see Algebraic Stacks, Lemma \ref{algebraic-lemma-map-space-into-stack}) which is an equivalence because $U \to \mathcal{X}$ is surjective, flat, and locally of finite presentation, see Algebraic Stacks, Remark \ref{algebraic-remark-flat-fp-presentation}. Thus we may assume that $\mathcal{X} = [U/R]$ where $(U, R, s, t, c)$ is a groupoid in algebraic spaces such that $s, t : R \to U$ are surjective, flat, locally of finite presentation, and locally quasi-finite. Set $$ U' = \coprod\nolimits_{u \in U\text{ lying over }x} \Spec(\kappa(u)). $$ The canonical morphism $U' \to U$ is a monomorphism. Let $$ R' = U' \times_\mathcal{X} U' = R \times_{(U \times U)} (U' \times U') $$ Because $U' \to U$ is a monomorphism we see that both projections $s', t' : R' \to U'$ factor as a monomorphism followed by a locally quasi-finite morphism. Hence, as $U'$ is a disjoint union of spectra of fields, using Spaces over Fields, Lemma \ref{spaces-over-fields-lemma-mono-towards-locally-quasi-finite-over-field} we conclude that the morphisms $s', t' : R' \to U'$ are locally quasi-finite. Again since $U'$ is a disjoint union of spectra of fields, the morphisms $s', t'$ are also flat. Finally, $s', t'$ locally quasi-finite implies $s', t'$ locally of finite type, hence $s', t'$ locally of finite presentation (because $U'$ is a disjoint union of spectra of fields in particular locally Noetherian, so that Morphisms of Spaces, Lemma \ref{spaces-morphisms-lemma-noetherian-finite-type-finite-presentation} applies). Hence $\mathcal{Z} = [U'/R']$ is an algebraic stack by Criteria for Representability, Theorem \ref{criteria-theorem-flat-groupoid-gives-algebraic-stack}. As $R'$ is the restriction of $R$ by $U' \to U$ we see $\mathcal{Z} \to \mathcal{X}$ is a monomorphism by Groupoids in Spaces, Lemma \ref{spaces-groupoids-lemma-quotient-stack-restrict} and Properties of Stacks, Lemma \ref{stacks-properties-lemma-monomorphism}. Since $\mathcal{Z} \to \mathcal{X}$ is a monomorphism we see that $|\mathcal{Z}| \to |\mathcal{X}|$ is injective, see Properties of Stacks, Lemma \ref{stacks-properties-lemma-monomorphism-injective-points}. By Properties of Stacks, Lemma \ref{stacks-properties-lemma-points-cartesian} we see that $$ |U'| = |\mathcal{Z} \times_\mathcal{X} U'| \longrightarrow |\mathcal{Z}| \times_{|\mathcal{X}|} |U'| $$ is surjective which implies (by our choice of $U'$) that $|\mathcal{Z}| \to |\mathcal{X}|$ has image $\{x\}$. We conclude that $|\mathcal{Z}|$ is a singleton. Finally, by construction $U'$ is locally Noetherian and reduced, i.e., $\mathcal{Z}$ is reduced and locally Noetherian. This means that the essential image of $\mathcal{Z} \to \mathcal{X}$ is the residual gerbe of $\mathcal{X}$ at $x$, see Properties of Stacks, Lemma \ref{stacks-properties-lemma-residual-gerbe-unique}.