  We study the thermodynamics of a one-dimensional attractive Fermi gas (the
Gaudin-Yang model) with spin imbalance. The exact solution has been known from
the thermodynamic Bethe ansatz for decades, but it involves an infinite number
of coupled nonlinear integral equations whose physics is difficult to extract.
Here the solution is analytically reduced to a simple, powerful set of four
algebraic equations. The simplified equations become universal and exact in the
experimental regime of strong interaction and relatively low temperature. Using
the new formulation, we discuss the qualitative features of finite-temperature
crossover and make quantitative predictions on the density profiles in traps.
We propose a practical two-stage scheme to achieve accurate thermometry for a
trapped spin-imbalanced Fermi gas.
