What does multiplication of an absolute value by a negative factor really mean?

Question

I was wondering, if -2*|-5| is actually -10, why are we also taught that vertical bars used to denote absolute value behave just like parentheses: they are grouping symbols and values can be distributed over them, like -2*((-2)+(-3)) = 10 because (-2)*(-2)+(-2)*(-3) = 4+6. So, is -2*|(-2)+(-3)| = 10 or -10? Because we know that -2*|(-2)+(-3)| = |(-2)*(-2)+(-2)(-3)| = |4+6| = 10, and yet we also know that -2*|(-2)+(-3)| = -2*|-5| = -2*5 = -10. Which one is it?

I am confused, and would appreciate your help.

Thank you

Answer 1

$-2*|(-2)+(-3)| = |(-2)*(-2)+(-2)(-3)|$

This equality is not true. Absolute values are not just a set of parentheses, but they also represent another operation of, in some sense, removing the sign from a number. This means that properties that hold for one thing may not hold for another.

In this case, the rule you'd want to use is $$ |xy| = |x| \cdot |y| $$ but this doesn't give you much to work with for your example since it lets you, at best, claim $$ -2 \cdot |(-2) + (-3)| = -1 \cdot \Big| 2 \big( (-2) + (-3) \big) \Big| $$ The negative can't go inside the absolute values. Otherwise you end up with nonsense like $$ -1 \cdot |x| = |-x| = |x| = |1 \cdot x| = 1 \cdot |x| $$ which, if $x \ne 0$, shows $-1 = 1$.

Answer 2

Absolute values do not behave like parentheses.

You can think of them as a process whereby you input a number on an absolute value and one of the two happens:

-If your number is positive the output is that exact number.
-If you input a negative number, the output is that number but with a preceding negative sign, so that the output is again positive.

If you are familiar with functions, the absolute value is a function, say $A(x), x\in \mathbb{R}$:

$$A(x)=\begin{cases} x \text{, if x}\ge0\\-x \text{, if x}\lt0 \end{cases}$$

It is clear that its range is $[0,+\infty)$.

You wouldn't say for an arbitrary function that $mA(x)=A(mx)$ so why make the claim for the absolute value?

As PrincessEev points out, the rule is $|x||y|=|xy|$. We can say though that $x|y|=|xy|$ if $x\ge0$.