Question

The number of integral value(s) of $x$ satisfying the inequality $\sqrt{(x-3)(2-x)}<\sqrt{4x^2+12x+11},$ is

Answer 1

$\sqrt{(x-3)(2-x)}<\sqrt{4x^2+12x+11}$
For the square root to exist,
$(x-3)(2-x)\ge 0\Rightarrow (x-3)(x-2)\le 0\Rightarrow x\in [2,3]$

$4x^2+12x+11\ge 0\Rightarrow 4x^2+12x+11\ge 0\Rightarrow D=144-4\times 44<0\therefore 4x^2+12x+11>0\forall x\in R$
So, $x\in [2,3]$

Now, squaring on both sides
$-x^2+5x-6<4x^2+12x+11\Rightarrow 5x^2+7x+17>0D=49-4\times 5\times 17<0\therefore 5x^2+7x+17>0\forall x\in R$

Therefore, $x\in [2,3]$
There are $2$ integral values in the range of $x$.