Having aspects of a societies’ legislative and social values of limits generates apathy among the electorate and leads to decreased democratic participation.
If citizens cannot modify and amend any and all of society’s principal values as they see fit, there is a reduced incentive to participate in the democratic process. Healthy public debate is replaced by apathy and indifference and society becomes democratically and intellectually impoverished. 
This generally isn’t true. The mere fact that some constitutional elements are unamendable makes them the target of intense public debate. The fact that all Americans are guaranteed the right to bear arms is guaranteed in its unamendable constitution (the extent to which the US Constitution is unamendable is contentious yet beyond the scope of this question) has not meant that discussions surrounding gun rights have vanished and the population has become indifferent. The opposite has occurred. Public debate intensifies and the constitutional element becomes a focal point. 
[P1] Populations become less democratically engaged when they have limited control over legislation. [P2] Unamendable constitutions reduce voters' control over legislation. [P3] Therefore, unamendable constitutions reduced democratic participation and are not compatible with free democracies.
Rejecting the premises
[Rejecting P2] Unamendable provisions in constitutions become the focal point of democratic debate, not a source of apathy.