Assuming the rent increase was valid (proper advance notice, amount does not exceed local guidelines–as in City of San Diego, etc.), then relative should immediately cash proferred check and give 3-day notice to pay remaining due rent.
In general, never turn down offered rent, since it lessens the financial hole landlord is in if this develops into a pissing contest in court. At this point the tenant seems only to be $200 short, which is not a big burden for them to come up with.
However, if tenant has not complied with rent increase in the past and landlord has not been vigilant in enforcing the increase, then landlord has implicitly accepted old rent as valid.
BTW, a hefty rent increase is not a bad way to get rid of a troublesome or expensive-to-keep tenant. Puts the option to move on the tenant and fairly imposes the appropriate burden on them if they chose to stay. Obviously, first be sure you are a good landlord in every respect.
