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I am having a little bit of problem with an inequality with nested absolute values:

$$|z^2-1| \ge |z+|1-z^2||$$

I've tried solving it by making three cases, $z\ge1$, $z\le-1$ and $z$ between $1$ and $-1$ and thus getting rid of absolute values for $z^2-1$ and $1-z^1$, and I am only left with 1 absolute value. But solutions at the end are not what they should be based on the graph. Here, $z$ is real, and WolframAlpha gives this solution.

What I am doing wrong?

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    As you wrote it the inequality is false. Take for example $$z=4\Longrightarrow |4^2-1|=15\,\,,\,\,|4+|1-4^2||=|4+15|=19$$ Check carefully what is what you *really* want to prove and write it down with LaTeX (go to FAQ section), to make it clear2012-11-26
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    @DonAntonio: I think OP is trying to find the $z$ for which the inequality is true.2012-11-26
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    Well, that'd definitely make more sense. I'm getting sloppy in guessing posters' intentions...2012-11-26

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