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Dear Taechers,

1. If your car is out of order in an accident and you patch it up with whatever is at hand.

- What does "at hand" mean here?

2. It's very difficult when it comes to trying to talk someone out of killing themselve.

- "talk someone out of doing sth" means "prevent someone from doing sth", right?

3. I think the most you can do is keep plugging away just like you have been with him.

- What does "plug away" mean here?

4. I think he's at a tough point in his life.

- "tough point" here means "tough period", right?

Thanks very much to Teachers,

Stevenukd.
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3. Plugging away is an uncommon idiom. It's tough to tell from the context, but "repeatedly trying" should fit most uses.
Comments  
1. Things at hand are the ones which are currently available to you, which are within your reach.

2. «Talk out» — Prevent from something, only by means of discussion. «Talk into» has the opposite meaning — to persuade somebody to do something.

3. Don't know

4. You are right: a tough point (=moment) is (of course) part of a tough period.
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 Vorpar's reply was promoted to an answer.
2.
"talk someone out of doing something" means to convince them not to do it.

3.
The idiom 'keep plugging away' (verb = to plug away) is common is Australia.

http://idioms.thefreedictionary.com/plug+away

The original sentence means: 'I think the most you can do is to be persistent just like you have been with him.'

4.
'I think he's at a tough point in his life.'
'Tough point' is more specific than 'tough period'.

For 'tough period' you would say 'I think he's going through a tough period (or tough stage) in his life.'

Often you would also add 'at the moment' or 'right now' to the end of these sentences.