What is "to work out"

Work \Work\ (w[^u]rk), v. t.

1. To labor or operate upon; to give exertion and effort to; to prepare for use, or to utilize, by labor.

   He could have told them of two or three gold mines, and a silver mine, and given the reason why they forbare to work them at that time.
   --Sir W. Raleigh.

2. To produce or form by labor; to bring forth by exertion or toil; to accomplish; to originate; to effect; as, to work wood or iron into a form desired, or into a utensil; to work cotton or wool into cloth.

   Each herb he knew, that works or good or ill.
   --Harte.

3. To produce by slow degrees, or as if laboriously; to bring gradually into any state by action or motion. ``Sidelong he works his way.''
   --Milton.

   So the pure, limpid stream, when foul with stains Of rushing torrents and descending rains, Works itself clear, and as it runs, refines, Till by degrees the floating mirror shines.
   --Addison.

4. To influence by acting upon; to prevail upon; to manage; to lead. ``Work your royal father to his ruin.''
   --Philips.

5. To form with a needle and thread or yarn; especially, to embroider; as, to work muslin.

6. To set in motion or action; to direct the action of; to keep at work; to govern; to manage; as, to work a machine.

   Knowledge in building and working ships.
   --Arbuthnot.

   Now, Marcus, thy virtue's the proof; Put forth thy utmost strength, work every nerve.
   --Addison.

   The mariners all 'gan work the ropes, Where they were wont to do.
   --Coleridge.

7. To cause to ferment, as liquor. To work a passage (Naut.), to pay for a passage by doing work. To work double tides (Naut.), to perform the labor of three days in two; -- a phrase which alludes to a practice of working by the night tide as well as by the day. To work in, to insert, introduce, mingle, or interweave by labor or skill. To work into, to force, urge, or insinuate into; as, to work one's self into favor or confidence. To work off, to remove gradually, as by labor, or a gradual process; as, beer works off impurities in fermenting. To work out.

   1. To effect by labor and exertion. ``Work out your own salvation with fear and trembling.''
      --Phil. ii. 12.

   2. To erase; to efface. [R.]

      Tears of joy for your returning spilt, Work out and expiate our former guilt.
      --Dryden.

   3. To solve, as a problem.

   4. To exhaust, as a mine, by working. To work up.

      1. To raise; to excite; to stir up; as, to work up the passions to rage.

         The sun, that rolls his chariot o'er their heads, Works up more fire and color in their cheeks.
         --Addison.

      2. To expend in any work, as materials; as, they have worked up all the stock.

      3. (Naut.) To make over or into something else, as yarns drawn from old rigging, made into spun yarn, foxes, sennit, and the like; also, to keep constantly at work upon needless matters, as a crew in order to punish them.
         --R. H. Dana, Jr.