In this sentence, two independent clauses are being joined by the coordinate conjunction "for." When this occurs, a comma must come before the coordinate conjunction. The correct answer choice is "he was trying to solve the problem, for his boss promised a bonus to the first person that could do it."

The phrase "which was open to the community" is an example of a relative clause, which gives more information about its antecedent: library. Any such phrase that occurs in the middle of a sentence must be introduced and completed with a comma. The correct answer choice is, "The school's library, which was open to the community, proved to be a valuable resource."

The phrase "which was supposed to revolutionize the industry" is an example of a relative clause, which gives more information about its antecedent: product. Any such phrase that occurs in the middle of a sentence must be introduced and completed with a comma. The correct answer choice is "people were still confused about what the new product, which was supposed to revolutionize the industry, actually did."

This sentence does not contain a comma where necessary. The phrase "While the rumors of the company's new product were all true" is an example of an introductory phrase, explaining information necessary to the context of the sentence, but outside of the main sentence, grammatically. These types of phrases must be followed by a comma in order to separate them from the rest of the sentence. The correct answer choice is "While the rumors of the company's new product were all true, there were many details."

The phrase "which was supposed to make their jobs easier" is an example of a relative clause, which gives more information about its antecedent: "machinery." Any such phrase that occurs in the middle of a sentence must be introduced and completed with a comma. The correct answer choice is "They were investing in new machinery, which was supposed to make their jobs easier, but cost much money."

The phrase "Due to the company's budget problems" is an example of an introductory phrase, explaining information necessary to the context of the sentence, but outside of the main sentence, grammatically. These types of phrases must be followed by a comma in order to separate them from the rest of the sentence. The correct answer choice is "Due to the company's budget problems, management started a round of layoffs. "

This sentence omits a comma where one is needed. A comma should always appear after the day of the month in a date. The correct answer is, "The kids were planning to have a parade on September 16, 2011." The other sentences are wrong because the commas are in the wrong place(s).

This sentence needs a comma after the word "days." The phrase "On nice days" is an example of an introductory phrase, explaining information necessary to the context of the sentence, but outside of the main sentence, grammatically. These types of phrases must be followed by a comma in order to separate them from the rest of the sentence. The correct answer choice is, "On nice days, Brandon liked."

The phrase "While answering questions" is an example of an introductory phrase, explaining information necessary to the context of the sentence, but outside of the main sentence, grammatically. These types of phrases must be followed by a comma in order to separate them from the rest of the sentence. The correct answer choice is, "While answering questions, Mathew was texting."

This sentence misuses a comma. While the conjunction "and" is present, there is no need for a comma because we do not yet have a list (three or more) of items. The comma should be omitted. Any other punctuation elsewhere is inappropriate since it would disturb the meaning of the sentence and does not belong, grammatically. The correct choice is, "collecting headphones and instruments."