# |x-|4-x||-2x=4 ??

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sciencesolve | Teacher | (Level 3) Educator Emeritus

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You should use the absolute value definition such that:

`|x| = {(x, x>=0),(-x, x<0):}`

Hence, considering the definition above, you should solve two equations such that:

`x - |4-x| = 4 + 2x`

`x - |4-x| = -4- 2x`

Solving the first equation yields:

`x - 4 - 2x = |4 - x|`

`-x - 4 = |4 - x|`

You should use the definition of absolute value such that:

`4 - x = x + 4 => 2x = 0 => x = 0`

`4 - x = -x - 4 => 4 = -4` invalid

Solving the second equation yields:

`x - |4-x| = -4 - 2x => x + 2x + 4 = |4 - x| => 3x + 4 = |4 - x|`

You should use the definition of absolute value such that:

`4 - x = 3x + 4 if 4 - x >=0 => x=< 4`

`4x = 0 => x = 0 =< 4`

`4 - x = -3x - 4 if 4 - x < 0 => x > 4 => x in (4 ,oo)`

`2x = -8 => x = -4 !in (4,oo)`

Hence, evaluating the solution to the equation yields x = 0.

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