Question

Find the range of $x$ for which $[4-\frac{x}{2}]=-1$, where $[.]$ denotes GIF.

Answer 1

Given: $[4-\frac{x}{2}]=-1$

Apply the property $\left[x\right]\le x<\left[x\right]+1$

$\Rightarrow [4-\frac{x}{2}]\le 4-\frac{x}{2}<[4-\frac{x}{2}]+1$
$\Rightarrow -1\le 4-\frac{x}{2}<-1+1$
$\Rightarrow -1-4\le -\frac{x}{2}<-4$
$\Rightarrow -5\le -\frac{x}{2}<-4$

Multiply by $-2$ and reverse the inequality sign.
$\Rightarrow -5(-2)\ge x>-4(-2)$
​​​​​​​$\Rightarrow 10\ge x>8$
​​​​​​​​​​​​​​$\Rightarrow 8<x\le 10$
​​​​​​​​​​​​​​​​​​​​​$\Rightarrow x\in (8,10]$