Mathematics - Quadratic Equation Question with Solution | TestHub

Given, ${\left|x\right|}^2-7\left|x\right|+9\le 0$

$\Rightarrow \left|x\right|\le \frac{7\pm \sqrt{13}}{2}$

$\Rightarrow \left|x\right|\in \left[\frac{7-\sqrt{13}}{2},\frac{7+\sqrt{13}}{2}\right]$

Also given $x\in \text{Integers}$, so $x$ can be $\pm 2, \pm 3, \pm 4, \pm 5$

Now out of these values of $x$ only $3, -4, -5$ will satisfy $S=\frac{\left|x+3\right|-1}{\left|x\right|-2}\ge 0$

So, $S\cap T\in \left\{3, -4, -5\right\}$

So, $n\left(S\cap T\right)=3$