Is $±x$ the same as $ |x|$?

Question

When solving a quadratic equation, the ± before the square root denotes "plus or minus that value." When solving absolute value equations, you try to solve using the positive and negative versions of the value inside. Does this mean that $y=|x|$ could be rewritten as $y=±x$?

Answer 1

$|x|$ is defined to be $x$ if $x\ge 0$ or $-x$ if $x<0$, so for every number $x$ there is a well-defined number $|x|$. $|\cdot|$ is, in other words, a function.

On the other hand, $\pm$ is not a function, it is a shorthand to avoid writing a long expression twice if the only modification you are making is converting the sign $+$ to $-$. For example, if you say: $y=\sqrt{x\pm\sqrt{x^2-1}}$, what you are really saying is:

$$y=\sqrt{x+\sqrt{x^2-1}}\text{ or }y=\sqrt{x-\sqrt{x^2-1}}$$

So, for example, "$y=\pm x$" means "$y=x\text{ or }y=-x$" - no more and no less. This is equivalent to $|y|=|x|$. In general, this is not equivalent with $y=|x|$ (because in general $y=|y|$ does not hold), but in some cases it may be equivalent if, from the context, you can somehow independently conclude that $y\ge 0$.

Answer 2

You're confusing implication with bi-implication. It is the case that $$y=|x| \implies y = \pm x,$$ but they are not equivalent statements.