The number of integral value(s) of $x$ satisfying $| x | | x - 5 | = 6$ is

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Solution

$| x | | x - 5 | = 6$
$\Rightarrow | x (x - 5) | = 6$
$∵ | a | | b | = | a b |$
$\Rightarrow x (x - 5) = \pm 6$
$\Rightarrow x^{2} - 5 x = \pm 6$

Case 1:
$\Rightarrow x^{2} - 5 x = 6$
$\Rightarrow x^{2} - 5 x - 6 = 0$
$\Rightarrow x^{2} - 6 x + x - 6 = 0$
$\Rightarrow (x - 6) (x + 1) = 0$
$∴ x = - 1, 6$

Case 2:
$\Rightarrow x^{2} - 5 x = - 6$
$\Rightarrow x^{2} - 5 x + 6 = 0$
$\Rightarrow x^{2} - 2 x - 3 x + 6 = 0$
$\Rightarrow (x - 2) (x - 3) = 0$
$∴ x = 2, 3$

Hence, the number of integral solutions of the equation is $4$ .

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