First, a little reminder. In Qing Liu's Book on algebraic Curves, algebraic varieties are defined as
Let $k$ be a field. An affine variety over $k$ is the affine scheme associated to a finitely generated algebra over $k$. An algebraic variety over $k$ is a $k$-scheme $X$ such that there exists a covering by a finite number of affine open subschemes $X_i$ which are affine varieties over $k$. A projective variety over $k$ is a projective scheme over $k$.
a little later, he describes the well-known example (Example 3.3.8) of the line with a doubled point at the origin (see also Hartshorne, II.2.3.6):
$$ -------:------- $$
Now as I understand it, this line is not separated and should therefore not be an algebraic variety. I just don't see where Liu's definition fails this example to be an algebraic variety.