Question

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

Answer 1

Please see the explanation.

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`