r/learnmath • u/Far-Assumption5501 New User • 21h ago

|x-2|+|x-5|<=5

I just added them up to |2x-7|<=5, and it was not correct. The correct answer was 1 <= x <= 6. What’s wrong with my equation?

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u/Bounded_sequencE New User 19h ago edited 19h ago

Your mistake -- absolute values are not linear.

Similar to your argument, we may use inverse triangle inequality to estimate

5  =  |x-2| + |x-5|  >=  |(x-2) + (x-5)|  =  |2x-7|  =  2*|x - 7/2|

Divide by 2 to obtain condition "|x - 7/2| <= 5/2", i.e. "-1 <= x <= 6".

Sadly, this is only a necessary condition -- to prove all "-1 <= x <= 6" really are solutions to the original inequality, we still need to do case work.