We consider large data problems for C1 solutions of the relativistic Euler equations. In the (1 + 1)-dimensional spacetime setting, if the initial data are strictly away from the vacuum, a key difficulty in considering the singularity formation is coming up with a way to obtain sharp enough control on the lower bound of the mass-energy density. For this reason, via an elaborate argument on a certain ODE inequality and introducing some key artificial (new) quantities, we provide one time-dependent lower bound of the mass-energy density of the (1+1)-dimensional relativistic Euler equations, which involves looking at the difference of the two Riemann invariants, along with certain weighted gradients of them. Ultimately, for C1 solutions with uniformly positive initial mass-energy density of the corresponding Cauchy problem, we give a necessary and sufficient condition for the singularity formation in finite time.