Prove that positive and negative numbers with an absolute value with equations with a variable in bars, too, having two solutions.

Question

I've read this and it's known that positive and negative numbers with an absolute value such as $|9|$ and $|-2|$ in an equation with a variable also in those bars on the other side have two solutions because the variable can be itself or the opposite of it using the same number from the absolute value bar to replace the variable. For example, for $|20|=|x|$, $x$ has two solutions, $20$ and $-20$ because the absolute values of those numbers are both $20$. Prove that this can happen.

Answer 1

This is indeed possible because of the way the absolute value was defined geometricaly. $|x|=|x-0|=d$ is interpreted as ---what are the numbers whose distance from 0 is d. From this interpretation and revisiting the real number line we see that we have $d$ and $-d$ as the answer to the question. The same argument holds whenever we have $|x-c|=d$ what are the numbers whose distance from $c$ is d. Note again that we have 2 answer here one at the right of $c$ and one at the left.