"Work out" has 9 definitions. Depending on the definition, "work out" can be inseparable or separable.
I have also included a few related expressions.
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Definitions
1. To engage in vigorous physical exercise.
["work out"]
"She has been working out almost everyday since the start of the New Year."
2. The total amount or result of a calculation.
["work out"]
"That works out to $13.59."
"The cost of the trip worked out cheaper than we had expected.
3. When something is successful - like when a situation or a plan has a good or desirable result.
["work out"]
"It'd be nice if things always worked out that way."
4. When a situation develops in a particular way.
Because a situation can develop in different ways, you'll need to describe the development.
["work out"]
"I hope your new job works out well."
"That advice you gave me worked out to be really helpful."
This definition is very similar to the previous definition but the emphasis here is on development, not success.
"I hope your new job works out well."
Refers to the development of the job - I am wishing it to be well.
"I hope your new job works out."
Refers to the outcome - that it will be successful.
5. Plan something in detail.
["work out"]
"I have to work out the budget before the deadline next Tuesday."
6. Decide or agree on something.
["work out"]
"We should work out a date for our next meeting."
7. To deal with a problem in a satisfactory way.
["work out"]
["work"] + [something/someone] + ["out"]
"You two need to work out your differences for the sake of the team."
"I need to work something out with management after what happened last week."
8. To solve a problem - "work (something) out."
["work out"]
["work"] + [something] + ["out"]
"I know we've run out of options, but we need to work out what to do."
9. To understand someone - "work (someone) out."
["work"] + [someone] + ["out"]
"Even though I've known him for so many years, sometimes I can't work him out."
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Related expressions
"(Things/everything) will work out" - a bad situation will eventually improve and have a desirable outcome.
"I'm so sorry to hear about your job. I'm sure things will work out and you'll get a better job in no time."
"How's that working out for you?" or "How'd that work out for you?" - a sarcastic and rhetorical question to point out that what someone is doing is not working out very well.
When you see this on TV shows or movies, it is usually set up as a joke. In real life, care needs to be taken about using this question, otherwise it will come across as condescending and rude.