[Math] Greatest integer function

algebra-precalculus

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? 🙂

Best 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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