Negative Numbers - GRE Quantitative Reasoning

Card 0 of 120

Question

Solve for :

Answer

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 :

$x = -\frac{94}{2}$

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).

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Question

Simplify $(7+x+3x^{4})-(x^{4}+x-2)$

Answer

The answer is $2x^{4}+9$

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

$(7+x+3x^{4})-(x^{4}+x-2)$

$(3x^{4}+x+7)+(-x^{4}-x+2)$

$2x^{4}+9$

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Question

Solve for :

Answer

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 .

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Question

Find the value of .

$-2x= -\frac{1}{3}$

Answer

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

$\left(-\frac{1}{3}\right) \div-2$ is the same as

$-\frac{1}{3} \cdot -\frac{1}{2}$ which is POSITIVE $\frac{1}{6}$

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Question

What is $\frac{-36}{-9}$ ?

Answer

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.

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Question

Solve for :

Answer

Begin by isolating your variable.

Subtract from both sides:

, or

Next, subtract from both sides:

, or

Then, divide both sides by :

$x=\frac{-10}{-5}$

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

$x=\frac{10}{5}$ or

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Question

Solve for :

Answer

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

$y = \frac{-45}{-3}$

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:

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Question

Solve for :

Answer

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

$p = \frac{-6}{2}$

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:

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Question

x, y and z are negative numbers.

A

---

x + y + z

B

---

xyz

Answer

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.

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Question

$-1\leq x\leq 1$ and $x\neq 0$

Quantity A:

Quantity B:

Answer

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.

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Question

Simplify:

$-(-6)^{2}(-1)-(-3)$

Answer

$-(-6)^{2}(-1)-(-3)$

$-(36)(-1)+(3)$

$(36)+(3)=39$

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

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Question

If and are both less than zero, which of the following is NOT possible?

Answer

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.

$2x \times y = \frac{1}{2}$ 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.

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Question

Simplify $(7+x+3x^{4})-(x^{4}+x-2)$

Answer

The answer is $2x^{4}+9$

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

$(7+x+3x^{4})-(x^{4}+x-2)$

$(3x^{4}+x+7)+(-x^{4}-x+2)$

$2x^{4}+9$

Compare your answer with the correct one above

Question

Solve for :

Answer

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 :

$x = -\frac{94}{2}$

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).

Compare your answer with the correct one above

Question

Solve for :

Answer

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 .

Compare your answer with the correct one above

Question

x, y and z are negative numbers.

A

---

x + y + z

B

---

xyz

Answer

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.

Compare your answer with the correct one above

Question

$-1\leq x\leq 1$ and $x\neq 0$

Quantity A:

Quantity B:

Answer

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.

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Question

Simplify:

$-(-6)^{2}(-1)-(-3)$

Answer

$-(-6)^{2}(-1)-(-3)$

$-(36)(-1)+(3)$

$(36)+(3)=39$

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

Compare your answer with the correct one above

Question

If and are both less than zero, which of the following is NOT possible?

Answer

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.

$2x \times y = \frac{1}{2}$ 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.

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Question

If and are integers such that $-9\leq a\leq 4$ and , what is the smallest possible value of ?

Answer

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

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