Solve the equation $| x - | 4 - x | | - 2 x = 4$

x = 0

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x = 1

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x = 2

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x = 3

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Solution

The correct option is A $x = 0$
Given equation
$| x - | 4 - x | | - 2 x = 4$ ....(1)
Case I: When $4 - x \geq 0$
$\Rightarrow x \leq 4$
So, in this case eqn (1) becomes
$| x - (4 - x) | - 2 x = 4$
$\Rightarrow | 2 x - 4 | - 2 x = 4$ ....(2)
Now, when $2 x - 4 \geq 0$ i.e $x \geq 2$
So,eqn (2) becomes
$2 x - 4 - 2 x = 4$
$\Rightarrow - 4 = 4$ which is absurd.
Hence, there is no solution for $x \geq 2$
Now, when $2 x - 4 < 0$ i.e $x < 2$
So, equation (2) becomes
$- (2 x - 4) - 2 x = 4$
$\Rightarrow - 4 x = 0$
$\Rightarrow x = 0$
Case II: When $4 - x < 0$
$| x + (4 - x) | - 2 x = 4$
$\Rightarrow - 2 x = 0$
$\Rightarrow x = 0$
Hence, $x = 0$ is the only solution to the given equation.

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