The number of solution(s) of the equation $| x - | 6 - x | | - 2 x = 6$ is

1

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0

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2

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infinite

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Solution

The correct option is A $1$
$| x - | 6 - x | | - 2 x = 6$
$\Rightarrow {\begin{matrix} | 2 x - 6 | - 2 x = 6, & if x \leq 6 6 - 2 x = 6, & if x > 6 \end{matrix}$

The $2$ nd case fails, because $x = 0$ is the only solution but condition is $x > 6$

Further, the $1$ st case becomes
${\begin{matrix} 2 x - 6 - 2 x = 6, & if 3 \leq x \leq 6 - 2 x + 6 - 2 x = 6, & if x < 3 \end{matrix}$
$\Rightarrow {\begin{matrix} - 6 = 6, & if 3 \leq x \leq 6 x = 0, & if x < 3 \end{matrix}$

So, from above condition, $x = 0$ is the only solution of the given equation.

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