How do you solve #abs(x-4)=abs(3x)#?

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Answer 1 · 2016-11-28T18:36:09

Please see the explanation.

The #|x - 4|# changes definition at between #x < 4 and x >= 4#:

#|x - 4| = x - 4; x >=4#
#|x - 4| = -x + 4;x < 4#

The #|3x|# changes definition at between #x < 0 and x >= 0#:

#|3x| = 3x; x >=0#
#|x - 4| = -3x;x < 0#

Therefore, the original equation has 3 different definitions corresponding to the 3 different domains:

The first doman:

#-x + 4 = -3x; x < 0#

#4 = -2x; x < 0#

#x = -2#

The second domain:

#-x + 4 = 3x; 0 <= x < 4#

#4 = 4x; 0 <= x < 4#

#x = 1#

The third domain:

#x - 4 = 3x; x >= 4#

#-4 = 2x; x >= 4#

No solution.

check x = 1 and x = -2

|1 - 4| = |3(1)|
|-2 -4| = |3(-2)|

3 = 3
6 = 6

This checks.

The answers are #x = 1 and x = -2#