He hangs on for his dear life.
I break into the jail.
I take off my hat.
I speed up in doing homework.
I slow down in doing homework.

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1.
Alice: He hang on for dear life.
Correct: He hangs on for his dear life.
Explanation:
*A singular subject takes a singular verb. (He hangs on...)
A plural subject takes a plural verb. (They hang on...)

2.
Alice: I break into sick.
Correct: I break into the office.
Explanation:
*Break into means you enter something illegally by force.

3.
Alice: I take of my hat.
Correct: I take off my hat. 
Explanation:
*Check your spelling. Take off

4.
Alice: I speed up doing homework.
Correct: I speed up in doing homework.
Explanation:
*Add in in your sentence.

5.
Alice: I slow down doing homework.
Correct: I slow down in doing homework.
Explanation:
*Add in in your sentence.

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Hello, Alice! Thank you for submitting your homework. Please read the following corrections and explanations carefully. If you have questions and clarifications, you can ask me in our class. Thank you!