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 $x = 1 and x = -2$

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