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If I have the equation $-5 = A + B$ and I decided to take the absolute value of both sides, would it evaluate to $\left|-5\right| = \left|A+B\right|$ or $\left|-5\right|=\left|A\right|+\left|B\right|$?

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    $\begingroup$ It would evaluate to $|-5| = |A+B|.$ $\endgroup$
    – xyz
    Commented Dec 20, 2022 at 16:26
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    $\begingroup$ The "sides" are $-5$ and $A+B$, so the absolute value fo both sides must be... $\endgroup$ Commented Dec 20, 2022 at 16:27
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    $\begingroup$ You can show that going from $-5=A+B$ to $|{-5}|=|A|+|B|$ is not valid, by finding an example of $A$ and $B$ where $-5=A+B$, but $|{-5}|\neq|A|+|B|$. For example, $A=1$ and $B=-6$ satisfy $A+B=-5$, but $|A| + |B| = |1| + |{-6}| = 1 + 6 = 7$, which is not the same as $|{-5}|=5$. $\endgroup$ Commented Dec 20, 2022 at 16:35
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    $\begingroup$ Remember what the $=$ sign means! It says that two expressions represent the same number. $\endgroup$
    – Karl
    Commented Dec 20, 2022 at 18:07

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Taking the absolute value of both sides of an equation $ E_1 = E_2 $ will evaluate to $ \left|E_1\right| = \left|E_2\right| $. In this case $ E_1 = -5 $ and $ E_2 = A + B $. From this follows that $ \left| -5 \right| = \left| A + B \right| \ne \left|A\right| + \left|B\right|$.

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