Question

If -2 <[x² -5]≤3 then the range of x is, where [.] represents greatest integer function.​

Answer 1

In the question, [x² - 5] is greatest integer function so it's an integer and it lies in the interval (-2, 3]. That means [x² - 5] can have any integer value in the interval (-2, 3], i.e.,

$\small\longrightarrow[x^2-5]\in\{-1,\ 0,\ 1,\ 2,\ 3\}$

We see that if a is an integer,

• $\small[x]=a\quad\iff\quad x\in[a,\ a+1)$

So, as we equate each integer in the set to [x² - 5], we get,

• $\small[x^2-5]=-1\quad\iff\quad x^2-5\in[-1,\ 0)$

• $\small[x^2-5]=0\quad\iff\quad x^2-5\in[0,\ 1)$

• $\small[x^2-5]=1\quad\iff\quad x^2-5\in[1,\ 2)$

• $\small[x^2-5]=2\quad\iff\quad x^2-5\in[2,\ 3)$

• $\small[x^2-5]=3\quad\iff\quad x^2-5\in[3,\ 4)$

The union of these 5 sets give range of x² - 5.

$\small\longrightarrow x^2-5\in\Big[[-1,\ 0)\cup[0,\ 1)\cup[1,\ 2)\cup[2,\ 3)\cup[3,\ 4)\Big]$

$\small\longrightarrow x^2-5\in[-1,\ 4)$

$\small\longrightarrow x^2\in[-1+5,\ 4+5)$

$\small\longrightarrow x^2\in[4,\ 9)$

$\small\text{ \longrightarrow\underline{\underline{x\in(-3,\ -2]\cup[2,\ 3)}} }$

This is the range of x.