Infinitesimal Translation Operator

In the book "Modern Quantum Mechanics" (by J.J. Sakurai and Jim Napolitano) page 44, infinitesimal Translation operator is given:
$mathscr{J}left(d mathbf{x}^{prime}right)left|mathbf{x}^{prime}rightrangle=left|mathbf{x}^{prime}+d mathbf{x}^{prime}rightrangle$ after this, book mentioned properties of the operator as being unitary and adding: $mathscr{J}left(d mathbf{x}^{prime prime}right) mathscr{J}left(d mathbf{x}^{prime}right)=mathscr{J}left(d mathbf{x}^{prime}+d mathbf{x}^{prime prime}right)$ where I have confused is that:

$mathbf{x} mathscr{J}left(d mathbf{x}^{prime}right)left|mathbf{x}^{prime}rightrangle=mathbf{x}left|mathbf{x}^{prime}+d mathbf{x}^{prime}rightrangle=left(mathbf{x}^{prime}+d mathbf{x}^{prime}right)left|mathbf{x}^{prime}+d mathbf{x}^{prime}rightrangle$

The left hand side of the equation is so weird to me because middle one is directly driven from the first equation but to find a meaning at the right hand side of the above equation, I thought it might be true because $d mathbf{x}^{prime}$ is infinitesimally small but where is $mathbf{x}^{prime}$came from because initially it was $mathbf{x}$ at the right hand side of the last equation.