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 $(-\infty ,-4]\cup (4,+\infty )$

If we solve it without concerning least integer and great integer then we get value of x somewhere in interval $(3,4)$
and if $x=(3,4)$ then $\left[x\right]=3$ and$\left(x\right)=4$
These values are not satisfying the equation so x has to be 4 at least. values of x has be 4 or greater. there is square on these term so we can also consider the negative values.
$|x|\ge 4$
$\Rightarrow x=(-\infty ,-4]\cup [4,+\infty )$