First of all, for warm-up, a practice Data Sufficiency question.

## The symbol

What does this symbol mean in math?

Technically, this symbol, typographically a dash, has three different meanings in mathematics, viz.:

a) a subtraction sign

b) a negative sign

c) an *opposite* sign

## The subtraction sign

When the dash appears between two terms — between two numbers (5 – 3), between two variables (x – y), between a number and a variable (x – 2), etc. — then it indicates the operation of subtraction. This is, undoubtedly, the very first meaning folks associate with the dash, because folks learned this way back in grammar school math. Furthermore, as soon as kids understand money and spending money, essentially they understand something about subtraction, so rare is the kid who doesn’t get subtraction early on.

## The negative sign

When the dash appears in front of a stand-alone number (–5, –2.7, , etc.), then it is a negative sign, denoting that the number in question is less than zero and to the left of zero on a standard number line. Folks tend to learn this idea relatively early on as well. Moreover, it’s easy to see how this meaning “blurs” into the subtraction-sign meaning, because after all, 8 + (–2) is just another way of saying 8 – 2. Both have a kind of “minus-making” meaning to them.

## The opposite sign

This is the one that can through folks into a tizzy. When you put the dash not in front of a stand-alone number but rather a stand-alone variable, then it is NOT a negative sign anymore. Rather, it is an opposite sign, which changes the sign of the variable to the opposite of whatever it was originally. If y is a positive, then –y is negative. BUT, if we know y is negative, then we know –y is positive.

Right there, that is precisely what wigs people out! Since before puberty, they were accustomed the dash having a universal “negatizing” effect, and yet, in this strange instance, when y is already negative, the dash in front of –y actually makes it positive. To some folks, this seems an unholy violation of everything they have ever learned about the sign! Technically, folks learn about the opposite sign somewhere in algebra, but it is seldom explained well there, setting folks up for this massive confusion when they encounter the opposite sign on, for example, the GMAT Quantitative section.

For example, the algebraic statement:

|y| = –y

is a sophisticated way of indicating that y ≤ 0. Conceivably, the GMAT could give you the former and expect you to deduce the latter.

## Practice

It may be that the foregoing discussion gave you some insight into the practice question at the top of this pages. Take another look at that before reading the solution below. Also, here’s a free practice question with some positive/negative variable issues.

2) http://gmat.magoosh.com/questions/301

## Practice Problem Solution

1) This is a difficult DS question. First of all, some folks might mistakenly think the prompt is already decided as it is written. After all, we know that the right side of the inequality,, is always positive. Some folks may mistakenly think that the left side of the inequality is always negative, but that would entail reading the dash incorrectly as a negative sign, rather than correctly as an opposite sign.

Statement #1 implies that y = ±1/5. (How do you know you have to take both the positive and negative square roots? See this GRE post.) Those two values imply different conclusions to the prompt. If y = +1/5, then –y/2 = –1/10, which is clearly *less than* . But, if y = –1/5, then –y/2 = +1/10, which happens to be *greater than* . The two different possible values imply different conclusions, which means no definitive answer to the prompt is possible. This statement, by itself, is **insufficient**.

Statement #2 is a fancy way of saying that y ≤ 0. If y = –5, then the inequality is false, but if y = –1/100, then –y/2 = +1/200, which is *greater than* , and the inequality is true. Two different possible values imply different conclusions, which means no definitive answer to the prompt is possible. This statement, by itself, is **insufficient**.

Statement combined: when we combine the restraints of both statements, we know that y can only have one value: y must equal –1/5, and this, by itself leads to a definite answer to the prompt. Thus, the combined statements are **sufficient**.

Answer = **C**.

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Hi Mike,

I do not understand why statment one in insufficient. If you devide both sides of the initial equation by y you get: -1/2 > y?

Statement 1 delivers that y could equal -1/5 and +1/5. Since both are greater values than -1/2 this would be sufficient to answer the question.

Could you help me to find where I made a mistake?

Thank you 🙂

Dear Oli,

I’m happy to respond. 🙂

Dividing by a variable is always a very risky activity, whether you are dealing with an equation or an inequality. It’s not like dividing by a number, say, 5. You can always divide either side of an equation or an inequality by 5, no worries, no bad consequences. Not so with a variable.

If y = 0, then dividing by y would violate the mathematical law and produce nonsense. That’s often a risk of dividing by a variable, though not here. Here, the risk is simply: is y positive or negative? We don’t know. If y is positive, then as you say, we divide and get -1/2 > y, which incidentally is a contradiction: y can’t be positive and less than -1/2. If y is negative, then when we divide, that should

reverse the direction of the inequality, which gives -1/2 < y. That's always a huge risk of dividing an inequality by a variable: if we don't know whether the variable's value is positive or negative, then we don't know whether the division will reverse the direction of the inequality.Does all this make sense?

Mike 🙂

Hello,

I read the post carefully but I’m sorry, I completely don’t understand why IyI means y ≤ 0. Please clarify somehow? Thanks.

Dear RT,

First of all, let’s be very clear here. The expression |y| by itself is just the absolute value of the variable y, and y could be either positive or negative, and the absolute value will make the output positive.

Similarly, the expression -y by itself just means the opposite of y. Whatever sign y has on its own, -y gives it the opposite sign: a positive y would be negative, and a negative y would become positive.

Neither one of those expressions, by themselves, in isolation, tells us that y has to be negative. In isolation, neither one allows us to figure out anything from y. In general, we can never figure anything about the value of a variable from an an expression alone. We always need an equation to figure out anything about the value of a variable.

Things change drastically when we take those two individual expressions together and put an equal sign between them. The equal sign is a mind-bogglingly powerful sign that many folks take for granted. When we create the equation |y|=(-y), then we have something quite significant. We know, from the left side, |y|, whatever the input is, the output has to be positive. Regardless of whether the variable y holds a positive or a negative number, the output of |y| must be positive. We know that. Well, the power of an equal sign is that whatever has to be true on one side has to be true on the other side as well. If the left side has to be positive, then the right side has to be positive as well. Well, if (-y), the opposite of y, is a positive number, that means that the original value of y must be a negative number, because the opposite of positive is negative. That’s why, not the individual expressions in isolation but the full equation, tells us that y must be a negative number. (Technically, y could also be zero, and that would work too.)

My friend, notice that the statement I made in the blog article was an equation (a mathematical statement with an equal sign), and the question you asked was about an expression (a mathematical statement without an equal sign). If you confuse these two, there is a ton in math that will never make sense. An equal sign is like the “verb” of a mathematical statement, and a person who confuses expression & equations is like a person who can’t tell the difference between a phrase and a full sentence (of course, that would be disastrous on GMAT SC!). There is a world of difference between what you can tell from an expression & what you can do with one vs. what you can tell from an equation & what you can do with one. This is precisely the sort of subtle distinction that you can never afford to overlook.

Does all this make sense?

Mike 🙂

Hey Mike,

Thanks for your speedy reply. You guys are great. I appreciate your trying to find another way to explain it to me, even though I am sure that to many, your above ‘mathematical discussion’ was perfectly clear. I read the entire page again, slooowly, and partly from being more fresh, partly from the round 2 of information intake (given that I looked it yesterday) and of course, partly because of you, it makes much more sense. I liked the ‘verb’ example, and how conversational all of Magoosh is.

So although my understanding here is still not perfect, it’s getting there- with this question anyway! (I hope/think)

: )

RT,

You are quite welcome, my friend. I’m glad you found this helpful. Best of luck to you!

Mike 🙂