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Question
Solve the inequality |x-2|≥|2x-3|

When trying to solve the above inequality, I considered 3 intervals, x<3/2, 3/2≤x<2 and x≥2.

When x<3/2, I understand why |2x-3|=-(2x-3) but I do not know how to determine |x-2|. Not sure whether its x-2 or -(x-2). Must we sketch the graph out in order to determine the sign? I saw your other answers and you just got them without working.

Answer
Hi Alex,
Its good but not always necessary to sketch out the graph. You should only realise, in this case, that the interval x<3/2 is contained in the interval x<2. Its actually inherent in your decision to choose those three intervals if you think about it. And its not even hard to see that any number less than 3/2 is also less than 2.
So, when x<3/2,
|x-2| = -(x-2)

And of course |x-2| = -(x-2) in the interval 3/2 < x < 2 as well.

Regards

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