Question

If $\left[x\right]=$ the greatest integer less than or equal to $x,$ and $\left(x\right)=$ the least integer greater than or equal to $x,$ and $[x{]}^2+(x{)}^2>25$ then $x$ belongs to

• A. $[3,4]$
• B. $(-\infty ,-4]$
• C. $(4,+\infty )$
• D. $(-\infty ,-4]\cup [4,+\infty )$

Answer 1

The correct option is D $(-\infty ,-4]\cup [4,+\infty )$

$[x{]}^2+(x{)}^2>25$

Say $x=n$, where $k\le n\le k+1$ ; $k\in Integers$

$\therefore [n{]}^2+(n{)}^2>25$ $\therefore \left[n\right]=k,\left(n\right)=k+1$

$\Rightarrow k^2+(k+1{)}^2>25$

$\Rightarrow k^2+k^2+1+2k>25$

$\Rightarrow 2k^2+2k-24>0$

$\Rightarrow k^2+k-12>0$

$\Rightarrow (k-3)(k+4)>0$

$k\in (-\infty ,-4)\cup (3,\infty )$

$\therefore x\in (-\infty ,-4]\cup [3+1,\infty )$

$\in (-\infty ,-4]\cup [4,\infty )$