GRE Question ($|x|$ vs $\sqrt{x^2}$)

Question

I just finished taking a Kaplan GRE Practice Test, and I encountered an interesting question with a very vague solution.

If you have taken the GRE, then you are probably familiar with the following question format. The question shows two quantities labeled as A and B, and you need to choose whether A is greater, B is greater, they are the same, or there is not enough information.

The question was the following

Quantity A: |x| Quantity B: $\sqrt{x^2}$

I answered that there is not enough information because the square root of $x$ has two solutions, the negative and positive root, whereas $|x|$ is always positive. Kaplan said that the solution is that they're the same (it looks like they are assuming that the only solution to $\sqrt{x^2}$ is the principal root.

Who is right and why?

Thanks!

Answer 1

It is most likely a mistyping on the exam's part or you, as I believe it should be $|x|$ vs. $\sqrt{x^2}$, as your statement doesn't work for $x=2$, as $|2|=2$ whereas $\sqrt{2}<2$.

In this case the quantities are equivalent, which you can see by always taking the positive solution to the square root, it must be that $\sqrt{x^2}=x$ if $x\geq 0$ and $\sqrt{x^2}=-x$ if $x<0$, which exactly defines $|x|$.

EDIT: For a more thorough explanation, let $y=\sqrt{x^2}$. Then clearly $x^2-y^2=0$. By expanding we get $(x-y)(x+y)=0$, so that either $y=x$ or $y=-x$, and we always take whichever one is positive.

EDIT 2: Here is a web cache from ETS's website (which writes the GRE), which states that, in their context, the symbol $\sqrt{\;\;}$ always is positive.

Answer 2

You are confused when you write they are "assuming that the only solution to $\sqrt{x^2}$ is the principal root." It makes no sense to speak of a "solution" here, because you have no equation.

By convention, the symbol $\sqrt{x}$ always denotes the nonnegative square root of $x$. We make that convention, among other reasons, so that $\sqrt{x}$ defines a function. In your problem, the symbol $\sqrt{x^2}$ stands for the nonnegative square root of $x^2$. So we must take the absolute value of $x$ to ensure the result is nonnegative. In short, the equation $\sqrt{x^2}=|x|$ is an identity. It holds for all real values of $x$.

(By contrast, the equation $\sqrt{x^2}=x$ is not an identity. It fails to be true for negative numbers. For example, $\sqrt{(-5)^2}\neq-5$. Again, that is because $\sqrt{(-5)^2}$ denotes the nonnegative square root of $(-5)^2=25$, which is 5.)

Many people abuse this notation. For example, many people write $\sqrt{4}=\pm2$. Such an equation is not only wrong but meaningless. $\sqrt{4}$ denotes a unique number. And in any case, no number can equal both $2$ and $-2$.

If you want to use the word solution in this context, you can say the following. The equation $x^2=c$, where $c$ is a positive constant, has two solutions, namely $\sqrt{c}$ and $-\sqrt{c}$. Regardless, $\sqrt{c}$ denotes a unique number.