One absolute value inside of another absolute value in the equation

Question

Let's say we have an equation:

$||x|-2| = |2|x|+4|$

How does one go solving it? Symbolab says that it currently doesn't support step by step explanation for this problem, so I would really appreciate if someone could do it!

Answer 1

You just have to use the definition of the absolute value:

$$|x| = \begin{cases} x, & \text{if $x$ is $\geq 0$} \\[2ex] -x, & \text{if $x$ is $< 0$} \end{cases}$$

Therefore, split the equation into

$$|x|-2 = |2|x|+4|$$ and

$$-(|x|-2) = |2|x|+4|$$

Now, keep doing that. Notice that there will be 4 equations and two of them are the same.

Answer 2

Either $|x|-2=2|x|+4$, or $|x|-2=-(2|x|+4)$.
Solve each of them to find $|x|$, and then find $x$.