Question

The sum of integral value(s) of $x$ satisfying the equation $\frac{|x+9|}{|x-5|}=8$ is

Answer 1

$\frac{|x+9|}{|x-5|}=8$
$\Rightarrow \left|\frac{x+9}{x-5}\right|=8$
$(\because \frac{|a|}{|b|}=\left|\frac{a}{b}\right|)$
$\Rightarrow \frac{x+9}{x-5}=\pm 8$

Case 1:
$\Rightarrow \frac{x+9}{x-5}=8$
$\Rightarrow x+9=8(x-5)$
$\Rightarrow 7x=49$
$\Rightarrow x=7$

Case 2:
$\Rightarrow \frac{x+9}{x-5}=-8$
$\Rightarrow x+9=-8(x-5)$
$\Rightarrow 9x=31$
$\Rightarrow x=\frac{31}{9}$

Hence, the sum of all integral value of $x$ is $7$.