Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

For this problem

$|2 - |x-2|| = 2$

I've found the values $x = -2$ and $x = 2$. However, an third solution was presented to me, which I can't seem to find by myself: $x = -6$.

Is this solution even valid? If so, how do I get to that solution?

share|cite|improve this question
up vote 4 down vote accepted

$-6$ isn’t a solution, as you can check by substituting it into the original equation, but $6$ is: $$|2-|6-2||=|2-4|=|-2|=2\;.$$

You can solve it in two steps.

First let $y=|x-2|$, and solve $|2-y|=2$: either $2-y=2$, in which case $y=0$, or $2-y=-2$, in which case $y=4$. Now substitute these possibilities into $y=|x-2|$.

If $y=0$, you have $|x-2|=0$, in which case $x=2$ is the only solution.

If $y=4$, you have $|x-2|=4$, so either $x-2=4$ and $x=6$, or $x-2=-4$ and $x=-2$.

Combining results, the three solutions are $x=-2$, $x=2$, and $x=6$.

share|cite|improve this answer
Thanks, this is actually an very elegant way to solve absolute values! – Miroslav Cetojevic May 25 '12 at 22:02

No, $x=-6$ is not a solution, since $|2-|{-6}-2||=|2-8|=6$. However, $x=6$ is a solution, since $|2-|6-2||=|2-4|=2$. In general, you can get all solutions by noting $$|2-|x-2||=a(2-|x-2|)=a(2-b(x-2))=2(a-b)-abx$$ where $a$ and $b$ are either $-1$ or $1$, find all solutions to $2=2(a-b)-abx$ and then check which ones actually work.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.