Grades 11-12 Video Solutions 2012-2012_11-12

Video Transcription

The set of solutions for the inequality absolute value of X plus absolute value of X minus 3 is greater than 3 is which of the following?  So these guys look pretty suspicious. X and X minus 3, they're only different, they only differ from each other by 3, which is this quantity right here.  So we can see that 3 is equal to the absolute value of X plus 3 minus X. This seems like a pretty strange way to write it.  But then by the triangle inequality, I can split this up into absolute value of X and then 3 minus X and then well this absolute value is the same one as the one that we have here.  So we see that 3 is always less than or equal to the quantity that they want. Now the question is when is it not actually equal to and when is it strictly greater?  Well, we know that the triangle inequality is only an equality when these two guys have the same sign  and otherwise it's a strict inequality. So that means either X is less than or equal to 0 and 3 minus X is less than or equal to 0. Well, we can see that that has no solutions  because if X is negative, then 3 minus X is positive.  They're never the same sign like that.  Otherwise, if X is positive and 3 minus X is also, you know, non-negative, then this gives us the solutions X is between 0 and 3. So this is when equality holds.  That means it's a strict inequality everywhere else which is this range right here.

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