Question: [This](https://matheducators.stackexchange.com/questions/5794/how-to-explain-that-a-negative-number-multiplied-by-a-negative-number-is-a-posit) recent question reminded me of a question: this year several students expressed concern about the expression $\sqrt{-x}$, on the grounds that it must be undefined because $-x$ is a negative number, and negative numbers don't have square roots. This is in the context of a calculus class for first-year college students, so these are students who ostensibly know algebra (and most of whom have taken a year of calculus already). (In the context of this class, characterizing negative numbers as not having square roots is correct; they know about complex numbers, and also know they're off-topic.)

My experience so far has been that these students can handle it after being reminded that it's possible that $x=-2$, so I can address it in an ad hoc way when it comes up, but I suspect it indicates a bigger conceptual gap. So my question is:

1. Why do students think this?
2. What algebra should I be reviewing to head this off before the specific case comes up?

Answer: 
I tutor many students whose teachers have been telling them to pronounce -x not as "negative x" but instead as "the opposite of x." I assume that this semantic change is aimed at instilling in students the idea that -x need not be a negative number.

You can ask questions such as "what is the opposite of negative 3" and "what is the opposite of 4" to get students used to this kind of thinking before you begin teaching about opposites of variables.
Pronouncing $-x$ as "negative $x$" leads to confusion because "negative" as an adjective means "less than zero". A red car is red; an even number is even; but negative $x$ might not be negative.

If math had evolved differently, we could have been using different symbolism today. Here's how the two symbolisms would translate:

$\begin{array}{rcl}
2 & : & \overset{\rightarrow}{2}\\
-2 & : & \overset{\leftarrow}{2}\\
-x & : & \overset{\rightleftharpoons}{x}\\
x-y & : & x + \overset{\rightleftharpoons}{y}
\end{array}$

You could show this to your students, tell them they can mentally translate between the two if it helps them, and pronounce "$-x$" / "$\overset{\rightleftharpoons}{x}$" as "$x$ flipped" to emphasize that it's what you get by flipping the number line.