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Question: Find all $x \in \mathbb R$ such that the inequality $4<|x+2|+ |x-1|<5$ is satisfied.

This is my attempt at solving the problem:

Case (i): If $x+2 \geq 0 $ and $ x-1\geq0$, then

$4<x+2+x-1<5$

$\implies 4<2x+1<5$

$\implies 3<2x<4$

$\implies \dfrac{3}{2}<x<2$

$\implies x\in (\dfrac{3}{2},2)$

Case (ii): If $x+2 < 0 $ and $ x-1<0$, then

$4<-x-2-x+1<5$

$\implies 4<-2x-1<5$

$\implies 5<-2x<6$

$\implies -3<x<\dfrac{-5}{2}$

$\implies x\in (-3,\dfrac{-5}{2})$

Case (iii): If $x+2 \geq 0$ and $x-1<0$, then

$4<x+2-x+1<5$

$\implies 4<3<5$, which is not true.

Case (iv): If $x+2<0$ and $x-1 \geq 0$, then $x<-2$ and $x \geq 1$, which is not possible.

So, from cases (i) and (ii), we have $x \in (-3,\dfrac{-5}{2}) \cup (\dfrac{3}{2},2).$

But $-2.75 \in (-3,\dfrac{-5}{2})$, which does not satisfy the inequality. Where did I go wrong in my calculation?

Also, what do I do if there are more than two absolute value functions? It would be quite tedious to check all cases. Is there an altogether different way to go about such problems involving the absolute value function without considering case-by-case?

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  • $\begingroup$ Why $-2.75$ does not satisfy the inequality? $\endgroup$ – user35603 Jul 27 '14 at 11:57
  • $\begingroup$ $x=-2.75$ does satisfy the inequality since $0.75+3.75=4.5$ and $4<4.5<5$, right? $\endgroup$ – MPW Jul 27 '14 at 12:01
  • $\begingroup$ For $x = -2.75$ you have: $\vert x +2\vert + \vert x - 1\vert = \vert -0.75\vert + \vert -3,75\vert = 0.75 + 3.75 = 4.5$ And $4 < 4.5 < 5$. As far as I know, you made no mistake in your reasoning. $\endgroup$ – Darth Geek Jul 27 '14 at 12:01
  • $\begingroup$ Oh wow, I made the silly error that $-2.75-1 = 1.75$. Sorry for all the trouble, and thank you! :) Can anyone answer the second part of my question, regarding an easier method for questions involving more than $2$ absolute value functions? $\endgroup$ – Train Heartnet Jul 27 '14 at 12:09
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It seems to me that your solution is completely correct. Look here for a diagram of the map $x \mapsto |x+2|+|x-1|$. It is clear that the solution set coincides with the set you found.

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Your answer is correct. For $x=-2.75$, $|-2.75+2|+|-|-2.75-1|=|-0.75|+|-3.75|=4.5$

So it does satisfy the inequality you came up with. Ironic though, your work was correct but you checking wasn't... funny how that goes.

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  • $\begingroup$ Funny indeed! This happens all the time with me, haha. :D $\endgroup$ – Train Heartnet Jul 27 '14 at 12:11
  • $\begingroup$ Actually, when I was plugging it in myself, I actually incorrectly evaluated it myself as 2.5 instead of 4.5... The only reason I noticed it was wrong is cause I evaluated it in Maple and it said it was 4.5 :P $\endgroup$ – Pauly B Jul 27 '14 at 12:14

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