Positives & Negatives

Perhaps no dichotomy is as important to Quantitative Comparisons as is the Positive/Negative distinction. For one thing, ETS can easily create a question about positives and negatives without having to use either word. But there are common clues. If you see any of the following clues in a QC question, ask yourself whether positive and negative numbers play a role:

1. Common information states that a variable is either greater than or less than 0:

   x < 0   means x is negative

   y > 0   means y is positive

2. The product of more than one variable is greater than or less than 0:

   pq > 0   means p and q have the same sign; they are either both positive or both negative

3. An expression that contains both a negative sign and an exponent:

   (−x)⁴   means (−x)⁴ is positive, since 4 is even

Greater Than or Less Than 0

These clues often mean that you can save time by making generalizations based on the signs of variables. For example:

x < 0
Quantity A		Quantity B
x − 2		− (x − 2)

You want not only to get this question right, but to get it right quickly. One option is to plug in numbers for x.

For instance, plug in −1 for x:

Quantity A		Quantity B
(−1) − 2 = −3		− ((−1) − 2) = 3

When x = −1, Quantity B is bigger.     

But do you know Quantity B will always be bigger? No, you would need to try other numbers for x, and that will be time-consuming.

Instead, see whether you can make a generalization about the sign of each of the quantities. If x is negative, can you say anything definite about the sign of x − 2? Yes, you can. A negative minus a positive will always be negative.

You can rewrite Quantity A:

Quantity A		Quantity B
Negative

Now you need to see whether you can make a generalization about Quantity B. Start with the expression inside the parentheses: x − 2. You know that x − 2 is always negative, so you can rewrite the expression as:

− (NEGATIVE)

What you have is a negative number inside the parentheses being multiplied by a negative:

(x − 2) is negative, so −(x − 2) is positive

You can rewrite Quantity B:

Quantity A		Quantity B
Negative		Positive

Instead of trying specific numbers, you made generalizations about the sign of each quantity. Any positive number is greater than any negative number, so Quantity B will always be greater. The correct answer is (B).

Strategy Tip:

If you are told the sign of a variable (e.g., x < 0), try to make a generalization about the sign of each quantity.

Product of Variables Greater Than or Less Than 0

In the last problem, you knew the sign of the variable. That will not always be the case:

xy > 0
Quantity A		Quantity B
x + y		0

This question is still about positives and negatives, but now it concerns the signs of both x and y. The common information is telling you something very important. There are two possible scenarios:

1. x and y are both positive.
2. x and y are both negative.

To find the answer, you need to test both scenarios. As in the last problem, you are testing not with specific numbers, but with the signs of the variables.

First, test the first scenario: x and y are both positive:

xy > 0
Quantity A		Quantity B
x + y Positive + Positive = Positive		0

If x and y are positive, Quantity A will always be positive, regardless of the values of x and y.

Now, test the second scenario: x and y are both negative.

xy > 0
Quantity A		Quantity B
x + y Negative + Negative = Negative		0

If x and y are negative, Quantity A will always be negative, regardless of the values of x and y. The correct answer is (D).

Strategy Tip:

When the product of more than one variable is either greater than or less than 0, consider all possible signs and test all possible scenarios.

If xy > 0, the two scenarios are:

1. x and y are both positive.
2. x and y are both negative.

If xy < 0, the two scenarios are:

1. x is positive and y is negative.
2. x is negative and y is positive.

You will have to test both scenarios to get the right answer consistently.

Exponents & Negatives

Another sign that you are dealing with positives and negatives is the combination of exponents and negative signs:

n is an integer.
Quantity A		Quantity B
(−3)2n		(−3)2n + 1

When negative numbers are raised to a power, they follow a pattern:

1. Negative numbers raised to odd powers are negative.
2. Negative numbers raised to even powers are positive.

You need to see if you can make a generalization about the sign of each quantity. Start with Quantity A: n is an integer, so 2n will always be even. The exponent will always be even, and a negative raised to an even power will always be positive:

n is an integer.
Quantity A		Quantity B
(Negative)Even = Positive		(−3)2n+1

Now, test Quantity B: 2n is always even, which means 2n + 1 will always be odd. A negative number raised to an odd power is negative:

n is an integer.
Quantity A		Quantity B
(Negative)Even = Positive		(Negative)Odd = Negative

The correct answer is (A).

Strategy Tips:

All of these problems have one thing in common: you can save time by figuring out whether each quantity is positive or negative.

Be on the lookout for these clues:

1. Common information states that a variable is greater than or less than 0
   (e.g., x > 0, p < 0).
2. Common information states the product of two variables is greater than or less than 0 (e.g., xy < 0).
3. An expression contains both an exponent and a negative sign (e.g., (−2)^x).

When problems contain both exponents and negative signs, try to make generalizations about the sign of each quantity:

1. A negative number raised to an odd power is negative.
2. A negative number raised to an even power is positive.