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I am struggling with this situation, I am from Australia, specifically Sydney, end of October I started a job but I had to relocate in Brisbane, I left my wife in Sydney because she has a stable job over there.

Basically after 3 weeks I received an offer from a company to be office located in Sydney with a much better salary than the current one and 30 minutes drive from my apartment, meantime in Brisbane, I am in a lease paid by me and to go to Sydney I have to pay my own flight tickets, I am having a hard time with expenses.

Now this company appears they want me under full permanent position and better salary, if I accept I would have to give 1 weeks notice to the current company since I am within the probation period which by law allows either party to end the employment at any time just giving 1 week notice period.

This dilemma is distressing me a lot due to my inflexibility of mind, and of course is creating financial and mental distress since I need to make the decision quickly or the offer will disappear, I managed to convince the second company to allow me to start January 2019.

I have drawn pros and cons and basically the second offer is far. far better just for the fact that I will be in Sydney with my family and saving cost will be exponential.

But, how to face the fact that I need to hand over the resignation letter, my idea is to present domestic problems which is the truth.

This is one of those cases where honesty is the best policy. Just explain to your manager that the long-distance relationship with your wife isn't working, you've had an offer back in Sydney, so here's your resignation letter.

They're probably not going to be ecstatic about it, but a probation period is there for both sides to work out if the job is the right fit. In this case it wasn't, so it's best for everyone just to move on.

The notice period of one week works both ways. If the company thought they don't want you they would give you one week's notice an be done with it.

I'm with Sandra K here - (see her comment). I don't think there is a dilemma here. From your post it's clear that the job in Brisbane is not working out and you want to move back to Sydney.

I think you should hand in the resignation letter. If you're asked why you're leaving just tell the truth.

You clearly have already emotionally taken a decision, so stick with it.

I know this might sound a bit "crude", but, business is business.
This employer has probably dealt already with long span sick leaves, bad interns, stubborn clients that don't want to pay... you will not be the first nor the last employee to leave, and he will know how to handle it.

You, and only you ( where you includes also your loved ones) is the only thing that should matter to you when making that decision.

You must assest the pros and cons, and in the end, decide what's best for you and your wife, no matter what anyone else ( in this case, a employer or project manager) in this world wants you to do.

And if that means taking the second job, so be it.

My (4 years ago, during my last job as "level 2 IT technician") Technical Director (project manager) told me, when i was in a similar situation:

"You owe nothing to this (or any) company but to do your job honestly, professionally and
efficiently ( to your best and beyond) while you work here. No more, no less.

Go on, take this
offer, and don't look back. We're all professionals, and it's your life and your career that should matter to you.".

I try to keep that honest advice from the person that was meant to "hold me back" as a golden rule.