When given the difference between numbers, do we use the absolute value?

This question is more about wording than computation.

The sum of the first number and the square of the second number is 18. The difference between the square of the second number and twice the first number is 12.

I know the first set of equations would be $$x+y^2=18$$ and $$y^2-2x=12$$, and we can get $$x=2$$ and $$y=2$$ or $$y=-2$$.

My question is about how to understand difference when making equations. I know it means subtraction but would you ever use absolute values when making the equation?

For example, if I said the difference between a number $$x$$ and $$7$$ is $$3$$, then wouldn't there be two answers? I would think of it as $$|x-7|=3$$ and then we could get two answers: $$x=10$$ and $$x=4$$. My rationale is that on the number line, the distance between either of my $$x$$ values and 7 is 3.

In the quoted question I began with, then I could get the numbers of $$x=10$$ and $$y=2\sqrt{2}$$ or $$y=-2\sqrt{2}$$ but that's only if I did $$|y^2-2x=12|$$ and I don't feel that's right.

• It depends on the writer (who may not be consistent). I might guess that because the question says "difference between the square of the second number and twice the first number" rather than "difference between twice the first number and the square of the second number", the intention was more likely to have been $y^2-2x=12$ than to be $|y^2-2x|=12$, but I could be wrong – Henry Jun 20 at 21:50

In the first case, $$y=\pm 4$$. The problem has two solutions, and the author did not specify "take the positive value" or, on occasion they will write along the lines of "the larger number minus 6 times the smaller number $$\ldots$$". I would not take the absolute value, that is not specified in the problem. They need to say "the absolute value of the difference is $$\ldots$$", otherwise you are solving a different problem. The two results come from the square root process.