Which of the following commands adds the directory /new/dir/ to the PATH environment variable?

$PATH= attempts to assign to the shell's expanded value of PATH rather than the variable name itself, which is invalid bash syntax.

Missing the export keyword means the updated PATH exists only in the current shell and is not inherited by child processes.

A space between the colon and $PATH breaks the PATH string into two words during assignment, preventing correct path concatenation.

Using $PATH on the left side of the assignment (export $PATH=) attempts to assign to the expanded string rather than the variable name, which is invalid syntax.

The export keyword makes the variable available to child processes. The assignment PATH=/new/dir:$PATH uses the bare variable name on the left side, places the new directory first, separates it from the current value with a colon, and expands the existing PATH using $PATH on the right - all required elements for correct PATH manipulation.