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I would like to understand the answer to the following problem.

Express the following with at least one less pair of absolute value signs: $$|(|a + b| - |a| - |b|)|$$

I know the answer is $|a| + |b| - |a+b|$

But I don't understand why this is this answer.

Please could someone explain how the answer was derived?

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  • $\begingroup$ Triangle inequality. $\endgroup$ – user142299 Jun 4 '14 at 18:23
  • $\begingroup$ Since you know the answer, you can try reverse engineering it by using the definition of absolute value: $|u| = u$ if $u \geq 0;$ $|u| = -u$ if $u \leq 0.$ $\endgroup$ – Dave L. Renfro Jun 4 '14 at 18:25
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The triangle inequality says that $|a|+|b|\ge |a+b|$. Result follows.

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Think about different cases: if $a \ge 0$ and $b \ge 0$, then $|a| = a$ and $|b| = b$, and $|a+b| = a+b$, so $|a|+|b|-|a+b| = a+b-(a+b) = 0$. What if $a < 0$ and $b \ge 0$? What if $a \ge 0$ and $b < 0$? What if both $a < 0$ and $b < 0$? In each case, what happens to the expression $|a|+|b|-|a+b|$?

Note to other readers: Yes, I am aware I am assuming that $a, b \in \mathbb{R}$ and I am aware of the triangle inequality; however, given the scope of the question and the likely background of the OP, my response is intended to address the question at an appropriate level.

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  • $\begingroup$ I have tried this with actual numbers and setting a = -1 and b = 2, then $|(|-1+2| -|-1| -|2|)| = 2$. I have tried this algebraically and with $a< 0$ and $b > 0$ is it correct to write $|(|a+b| -|a| - |b|)| = |((-a + b) - a -b)| = |-2a| = 2a$? I still can't make the link with how this helps get the solution though. Sorry for not understanding. $\endgroup$ – mikoyan Jun 4 '14 at 19:15

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