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"Determine which of the following are solutions of the equation $\lfloor x \rfloor = \lceil -x \rceil - 6$"

I understand there are two methods to finding a solution for the floor when $x$ is an integer and $x$ is not an integer.

Just wondering if someone could run through the steps in determining these solutions with me?

Thank you!

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2  
Hint: $\lceil -x\rceil=-\lfloor x\rfloor$ –  Raymond Manzoni Apr 7 '13 at 10:40

2 Answers 2

Hint: Write $x = n + r$, where $n = \lfloor x \rfloor$ is an integer, and $0 \le r < 1$ is possibly a fraction. Now,
$-x = -n - r$ so $\lceil -x \rceil = -n$. Can you use these in the equation?

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I agree with Macavity too.

Let $n\in (\Bbb Z>0)$.

If $0<x $ then $x=n+r$ such that $0 \le r <1 \Rightarrow [x]=n$ and $[-x]=-n-1 \Rightarrow n=-n-1-6 \Rightarrow 2n=-7$ does not have answer!

Else if $0>x$ then $x=-n+r$ and $0 \le r <1 \Rightarrow [x]=-n$ and $[-x]=n-1 \Rightarrow -n=n-1-6 \Rightarrow 2n=7 \Rightarrow n=\frac{7}{2}$ so does not have answer!

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I tried to improve your post using TeX (for better readability). Please check whether these edits did not unintentionally change the meaning of your post. –  A.P. Apr 7 '13 at 12:06
    
Thank you so much :) –  Somaye Apr 7 '13 at 12:12

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