We establish several Jackson type inequalities with explicit constants for spline approximation of functions defined on the real axis. The inequalities for the first modulus of continuity of odd derivatives are sharp. We also obtain inequalities for high-order moduli of continuity of a function itself. One of the inequalities for the second modulus of continuity is sharp. Up to the present paper no estimates for spline approximation on the axis in terms of high-order moduli of continuity, with constants written explicitly, were known.