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This is a homework puzzle so I'm not asking for the direct answer.

Find all numbers $x$ in $\Bbb R$ for:

$$[x+2] = 6[x] - 23$$

I haven't see greatest integer functions that have a scalar out the front nor two GIF in one function. Could someone please help me understand how to solve this? :)

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Sketch a graph? Or note that each side of the equation will always be an integer, so solve for $x$ an integer, and then consider the range of solutions around that. –  Mark Bennet Mar 15 '12 at 10:29
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1 Answer

I think it's long enough that I can add a full answer. By the first hint, you have that,

$$\begin{align}[x+2]\overset{1}{=}[x]+2&=6[x]-23\\5[x]&=25\\\ [x]&=5\\(2) \implies 5\le &x \lt6\end{align}$$

So, the solution is $$\boxed{5 \le x \lt6}$$


Hint:

  1. $[x+I]=[x]+I$ for $I$ an integer.
  2. $[x]=I \implies I\le x \lt I+1$ for $I$ an integer.
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$-1$ We don't know if the homework deadline passed or not. Also, we can't be sure that the OP has already solved the problem using the hints in the previous version.. –  user2468 Mar 15 '12 at 21:12
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Are you going to down vote because of that? –  user21436 Mar 15 '12 at 21:15
    
And BTW, I am not sure if I can take an answer that elaborates nothing more than my hints an hour later getting three upvotes for no reason. –  user21436 Mar 15 '12 at 21:17
    
@JD I would now edit to remove that answer. Would you retract the down vote? –  user21436 Mar 15 '12 at 21:25
    
I did not downvote or upvote anyone on this thread. I just wrote "-1" in my comment. Obviously someone else downvoted your answer! –  user2468 Mar 15 '12 at 21:28
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