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Question 1108270: If a presidential candidate is 42 votes ahead of his opponent before the votes for the state of California are added, what absolute value equation would represent the margin of votes between the candidate and his opponent after California’s 55 votes are cast?
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! here's a good reference that helps you to solve problems like these.
https://www.purplemath.com/modules/absineq.htm
the candidate is now ahead of his opponent by 42 votes.
the best he can do is gain all of the 55 votes, in which case he will have 97 votes more than his opponent.
the worst he can do is lose all of the 55 votes, in which case he will have 13 votes less than his opponent.
the number of votes he has more than his opponent will therefore be between -13 and 97.
if you let x equal the the number of votes he has more than his opponent, you get -13 <= x <= 97.
to put this in absolute value equation form, you need the left side of the inequality to be equal to the right side of the inequality.
take the difference between 97 and -13 to get 110.
divide this by 2 to get 55.
you want the left side of the inequality to be equal to -55 and the right side of the inequality to be equal to 55.
on the left side of the inequality, -55 - (-13) is equal to -42.
on the right side of the inequality 55 - 107 is equal to -42.
therefore, you subtract 42 from all sides of the equation to get:
-13 - 42 <= x - 42 <= 97 - 42
this results in -55 <= x - 42 <= 55
this can now be put into absolute equation form of |x - 42| <= 55
when you solve this absolute value equation, you get:
x - 42 <= 55 results in x <= 97
x - 42 >= -55 results in x >= -13
that's the same as -13 <= x <= 97, so you're good.
your solution is that |x - 42| <= 55
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