Express `f(x) = |1/(2x) - 2| - |1/(2x) + 2|` in the non-modulus form?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The function `f(x) = |1/(2x) - 2| - |1/(2x) + 2|`

`|x|` is equal to x if `x>=0` and it is equal to `-x` if `x < 0`

`f(x) = |1/(2x) - 2| - |1/(2x) + 2|`

If `x >= 1/4` , `f(x) = 1/(2x) - 2 - 1/(2x) - 2 = -4`

If `x <= -1/4` , `f(x) = -1/(2x) + 2 + 1/(2x) + 2 = 4`

If `-1/4 < x <1/4` , `f(x) = 2 - 1/(2x) - 1/(2x) - 2 = -1/x`

The function `f(x) = |1/(2x) - 2| - |1/(2x) + 2|` is the same as `f(x) = [[-4, x >= 1/4],[-1/x, 1/4 > x > -1/4],[4, x<= -1/4]]`

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sevenmustangs | Student, Undergraduate | (Level 1) eNoter

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Find separate solutions to the factors inside modulus

then 1/(2x)-2=0  => 1/(2x)=2 => x=1/4

and 1/(2x)+2=0 => 1/(2x)=-2 => x=-1/4

In number line -1/4 comes before 1/4 so when x<-1/4 both modulus will be negative

so for x<-1/4 = -(1/2x-2)-(-(1/2x+2)) = 4

 

for -1/4<x<1/4  

= -(1/2x-2)-(1/2x-2) = -1/x 

 

& for x>1/4  = (1/2x-2)-(1/2x-2) = -4




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