# Absolute value of an equation

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 $$\left|-5\right|=\left|A\right|+\left|B\right|$$?

• It would evaluate to $|-5| = |A+B|.$
– xyz
Commented Dec 20, 2022 at 16:26
• The "sides" are $-5$ and $A+B$, so the absolute value fo both sides must be... Commented Dec 20, 2022 at 16:27
• 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 $|{-5}|=5$. Commented Dec 20, 2022 at 16:35
• Remember what the $=$ sign means! It says that two expressions represent the same number.
– Karl
Commented Dec 20, 2022 at 18:07

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 $$\left| -5 \right| = \left| A + B \right| \ne \left|A\right| + \left|B\right|$$.