I'm Lucy, and I'd be happy to answer your questions today. I'm sorry to hear about your situation.
California law unfortunately only requires 30 days written notice to vacate a tenancy for tenants who have lived in a property less than one year. Cal. Civ. Code, Section 1946.1. The law doesn't give the tenant any additional time for not opening the letter when it's received - if that were the case, tenants would never open letters from their landlords and it would be nearly impossible to terminate a month-to-month lease. You said that you opened the notice on May 28 and the letter gives you until June 30 at the latest, so you still got 30 days notice.
Without a written lease agreement, the landlord's reasons for asking you to leave are irrelevant. Unless the eviction were based on you requesting repairs are your status as a member of a minority group, he's allowed to give you 30 days notice just because he wants to. If the notice says "At that latest, June 30," that's the date you need to be out. It is not legally relevant that he stated incorrect facts in the notice, because he doesn't have to state any facts at all. The only thing the notice needs is the date you're required to leave, at least 30 days after when they give you the notice.
Not paying rent adds an additional problem, because it means he can now serve you with a 3 day notice to pay or quit and go to court to evict you earlier than June 30. That means filing a Complaint, paying a sheriff to serve it on you (or having a friend hand it to you), going to trial, getting a judgment and a writ of execution, having it posted, and then having the sheriff forcibly remove you from the premises. When it gets to that point, the sheriff could place all of your belongings on the sidewalk and neither the state nor the landlord would be responsible if someone took them before you could move them to a secure location. Your landlord cannot do that if you pay rent through the end of the month. Court typically takes a few weeks, so chances are you wouldn't have a hearing date before June 30 at this point, but your landlord could report the late payment on your credit.
I'm sorry to say, the only options for a tenant being given a notice to terminate a month to month lease are to move out voluntarily or wait and see if the landlord evicts them. If he goes to court on July 1, the case could be over before the end of the month. Your rental history would show an eviction, and it could become more difficult to find a new place to rent in the future. The landlord can also seek a judgment for the unpaid rent at the same time, which would go on your credit. A judgment could also be used to garnish your wages, which means your employer would see it.
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