What are different integer values of $k$ between $1-9$ for which the equation $$|x-1|+|x-2|+|x+1|+|x+2|=4k$$,has no solutions. Now there are 24 different ways of having signs ie the equation after removing mod.solving these $24$ equations and then getting answer is very much time consuming(though I got the answer) So my main problem is how to deal with the mod sign and proceed with it or is there any shortcut.
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2Is $x$ a real number, or an integer ? What is a "mod sign" ? – Dietrich Burde Jun 04 '16 at 11:54
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@DietrichBurde $\mid \cdot \mid$ – snulty Jun 04 '16 at 12:03
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Dietrich Burde nothing is mentioned about x only given that k is an integer – Archis Welankar Jun 04 '16 at 12:20
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How do you get 24? Four two-way choices give 16 options. But even that is way too much! You just need to consider the 5 intervals which the real line is divided into by the points $\pm 1$ and $\pm 2$... – Hans Lundmark Jun 04 '16 at 13:33
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The left-hand side, call it $f(x)$, is a piecewise linear function, and convex, so the minimum value must be one of $f(-2)$, $f(-1)$, $f(1)$ and/or $f(2)$. There are no solutions if and only if $4k$ is less than that minimum.
Hans Lundmark
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The graph looks like this: http://www.wolframalpha.com/input/?i=plot+abs%28x+-+1%29+%2B+abs%28x+-+2%29+%2B+abs%28x+%2B+1%29+%2B+abs%28x+%2B+2%29, so $f(x)$ takes all values from 6 and upwards. So there is no solution iff $4k < 6$, i.e., $k<3$. (I'm assuming that $x$ is a real number. If $x$ has to be an integer, that should have been stated in the question. But in that case, it's also easy to read off from the graph what the answer should be.) – Hans Lundmark Jun 04 '16 at 15:15
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