Negative Numbers - GRE Quantitative Reasoning

To solve this problem, you need to get your variable isolated on one side of the equation:

Taking this step will elminate the on the side with :

Now divide by to solve for :

The important step here is to be able to add the negative numbers. When you add negative numbers, they create lower negative numbers (if you prefer to think about it another way, you can think of adding negative numbers as subtracting one of the negative numbers from the other).

To solve this problem, first you must add to both sides of the problem. This will yield a result on the right side of the equation of , because a negative number added to a negative number will create a lower number (i.e. further away from zero, and still negative). Then you divide both sides by two, and you are left with .

To solve for , divide each side of the equation by -2.

is the same as

which is POSITIVE

A negative number divided by a negative number always results in a positive number. divided by equals . Since the answer is positive, the answer cannot be or any other negative number.

Begin by isolating your variable.

Subtract from both sides:

, or

Next, subtract from both sides:

, or

Then, divide both sides by :

Recall that division of a negative by a negative gives you a positive, therefore:

or

To solve this equation, you need to isolate the variable on one side. We can accomplish this by dividing by on both sides:

Anytime you divide, if the signs are the same (i.e. two positive, or two negative), you'll get a positive result. If the signs are opposites (i.e. one positive, one negative) then you get a negative.

Both of the numbers here are negative, so we will have a positive result:

To solve, you need to isolate the variable. We first subtract then divide by :

When dividing, if the signs of the numbers are the same (i.e. both positive, or both negative), you yield a positive result. If the signs of the numbers are opposites (i.e. one of each), then you yield a negative result.

Therefore:

Recognize the rules of negative numbers: if two negative numbers are multiplied, the result is positive. However if three negative numbers are multiplied, the result is negative. As such, we know B must be negative.

Since there are no restrictions on the values of x, y and z beyond being negative, lets check low values and high values: if every value was -1, multiplying the values would equal -1 while adding them would equal -3. However, if every value was -5, multiplying them would equal -125 while adding them would equal a mere -15. As such, we would need additional information to determine whether A or B would be greater.

Imagine two different scenarios when x equals either extreme: –1 or 1. If x equals –1, then x squared equals 1 and x cubed equals –1 (a negative times a negative times a negative is a negative), and thus Quantity A is greater. The other scenario is when x equals 1: x squared equals 1 and x cubed also equals 1. In this scenario, the two quantities are equal. Because both scenarios are possible, the relationship cannot be determined without more information.

Remember, the product of two negatives is positive. Also note that subtracting a negative is equivalent to adding its absolute value.

This question tests your familiarity with the mathematical principles behind how negative numbers operate.

is possible because two negative numbers added together will always equal a negative number.

is possible because xy and -yx are inverses of each other, so they will combine to make 0.

is possible because you don't know what the values of x and y are. If y is sufficiently larger than x, then subtracting the negative number resulting from 2y (aka adding 2y) to the negative number 3x could be a positive number, including 5.

is possible because a negative (2x) times a negative (y) will always be positive.

Which, of course, means that is impossible, because a negative times a negative will never equal a negative.

To make as small as possible, let be as small as possible , and subtract the largest value of possible :

Since both quantities have an , you can ignore this variable, which will give you and for quantities A and B, respectively. Since and are both negative numbers, must be bigger than , which means that no matter what numbers they are, Quantity A must be bigger.

A problem like this one is very easy. All you need to do is manage your arithmetic well. Remember that when you subtract a negative number, this is the same as adding the positive of that number. Therefore, you can rewrite each of your quantities:

Quantity A:

Using a calculator, this comes out to be:

Quantity B:

Using a calculator, this comes out to be:

Therefore, quantity B is larger.

To solve this problem, you need to get your variable isolated on one side of the equation:

Taking this step will elminate the on the side with :

Now divide by to solve for :

The important step here is to be able to add the negative numbers. When you add negative numbers, they create lower negative numbers (if you prefer to think about it another way, you can think of adding negative numbers as subtracting one of the negative numbers from the other).

The answer is

Make sure to distribute negatives throughout the second half of the equation.

To solve this problem, first you must add to both sides of the problem. This will yield a result on the right side of the equation of , because a negative number added to a negative number will create a lower number (i.e. further away from zero, and still negative). Then you divide both sides by two, and you are left with .

To solve for , divide each side of the equation by -2.

is the same as

which is POSITIVE

A negative number divided by a negative number always results in a positive number. divided by equals . Since the answer is positive, the answer cannot be or any other negative number.