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

Nov 28, 2016

#### Explanation:

The $| x - 4 |$ changes definition at between $x < 4 \mathmr{and} x \ge 4$:

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

The $| 3 x |$ changes definition at between $x < 0 \mathmr{and} x \ge 0$:

|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 $x = 1 \mathmr{and} x = - 2$