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1. When To Use Commas

Use a comma to separate items in a series, including the next-to-last word in the series.

We develop solar thermal, wind, biomass, and photovoltaic energy technologies.

Use a comma to separate the parts of a compound sentence linked by a coordinating conjunction (such as "and," "but," "or," and "nor") when each part has its own subject and verb (unless they're very short).

I laughed at the unintentional joke, but she frowned.

Use commas to set off nonessential or nonrestrictive (parenthetical) words, phrases, and clauses from the rest of the sentence. In other words, the commas signal that the information between them is something extra and not essential to the meaning of the sentence.

The subsystem, which takes a day to install, will be delivered in two weeks.

Use commas to enclose the name of a state when it follows a city and a year when it follows the month and day.

The test systems in Gardner, Massachusetts, are performing well.
The next test sites will be in Golden, Colorado, and Chapel Hill, North Carolina.
On April 11, 1998, the committee members completed five of the six objectives.

2. When Not To Use Commas

Do not use a comma to separate compound subjects or compound verbs.

Theorists and nonspecialists alike agree on the importance of the discovery. (There is no comma between the two parts of this compound subject.)
The researchers rolled out the thin metal sheet and formed it into coils. (There is no comma between the two parts of this compound verb.)

Do not use commas to set off words or phrases that are restrictive (i.e., essential to the meaning of a sentence).

Only the sensors that were attached to the outer edge failed. (The words are essential to the meaning of the sentence.)
The system will work efficiently only if it includes storage. (The words are essential to the meaning.)

See also which and that.