Given an irrational slope $α$ and a real intercept $β$, the inhomogeneous linear approximation problem asks for integers $m$ for which $\norm{mα-β}$ is small compared with $\abs{m}$. Building on Ostrowski's numeration system, we give a self-contained treatment of its alternating (dual) counterpart, whose weights are the signed convergent denominators $(-1)^kq_k$, and show that with $n$ digits it represents exactly the integers of an explicit interval of length $q_n+q_{n-1}$, compared with $q_n$ for Ostrowski's system. We pair the absolute and the alternating integer systems with, respectively, Ostrowski's classical alternating expansion and an absolute expansion of real numbers in the linear approximation coefficients $θ_k=q_kα-p_k$. These expansions yield explicit algorithms that produce, for the one--sided and the two--sided (forward and backward) variants of the problem, sequences of approximates $A_n$ with $\abs{A_n}\le q_n$ and error at most $\abs{θ_{n-1}}$. For the one--sided variants we prove optimality: the $n$--th term is the best one--sided approximate within the $n$--th range of the alternating system.

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