# PSAT Math : Negative Numbers

## Example Questions

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### Example Question #1 : How To Add Negative Numbers

How many elements of the set  are less than  ?

Four

One

Three

None

Two

Four

Explanation:

The absolute value of a negative number can be calculated by simply removing the negative symbol. Therefore,

All four (negative) numbers in the set  are less than this positive number.

### Example Question #13 : Integers

a, b, c are integers.

abc < 0

ab > 0

bc > 0

Which of the following must be true?

b > 0

ac < 0

a + b < 0

a – b > 0

a > 0

a + b < 0

Explanation:

Let's reductively consider what this data tells us.

Consider each group (a,b,c) as a group of signs.

From abc < 0, we know that the following are possible:

(–, +, +), (+, –, +), (+, +, –), (–, –, –)

From ab > 0, we know that we must eliminate (–, +, +) and (+, –, +)

From bc > 0, we know that we must eliminate (+, +, –)

Therefore, any of our answers must hold for (–, –, –)

This eliminates immediately a > 0, b > 0

Likewise, it eliminates a – b > 0 because we do not know the relative sizes of a and b. This could therefore be positive or negative.

Finally, ac is a product of negatives and is therefore positive. Hence ac < 0 does not hold.

We are left with a + b < 0, which is true, for two negatives added must be negative.

### Example Question #61 : Integers

What is ?

45

Explanation:

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.

### Example Question #5 : Negative Numbers

Solve for :

Explanation:

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

### Example Question #2 : Negative Numbers

If  is a positive number, and  is also a positive number, what is a possible value for ?

Explanation:

Because is positive,  must be negative since the product of two negative numbers is positive.

Because  is also positive, must also be negative in order to produce a prositive product.

To check you answer, you can try plugging in any negative number for .

### Example Question #21 : Integers

, and  are all negative odd integers. Which of the following three expressions must be positive?

I)

II)

III)

I only

All of these

None of these

II only

III only

All of these

Explanation:

A negative integer raised to an integer power is positive if and only if the absolute value of the exponent is even. Since the sum or difference of two odd integers is always an even integer, this is the case in all three expressions. The correct response is all of these.

### Example Question #1 : Negative Numbers

is a positive integer;  and  are negative integers. Which of the following three expressions must be negative?

I)

II)

III)

None of I, II or III

I and II only

II and III only

I and III only

I, II and III

None of I, II or III

Explanation:

A negative integer raised to an integer power is positive if and only if the absolute value of the exponent is even; it is negative if and only if the absolute value iof the exponent is odd. Therefore, all three expressions have signs that are dependent on the odd/even parity of  and , which are not given in the problem.

The correct response is none of these.

### Example Question #23 : Integers

, and  are all negative numbers. Which of the following must be positive?

Explanation:

The key is knowing that a negative number raised to an odd power yields a negative result, and that a negative number raised to an even power yields a positive result.

and  are positive, yielding a positive dividend;  is a negative divisor; this result is negative.

:  and  are negative, yielding a positive dividend;  is a negative divisor; this result is negative.

:  is positive and  is negative, yielding a negative dividend;  is a positive divisor; this result is negative.

:  is negative and  is positive, yielding a negative dividend;  is a positive divisor; this result is negative.

is positive and  is negative, yielding a negative dividend;  is a negative divisor; this result is positive.

The correct choice is .

### Example Question #24 : Integers

and  are positive numbers;  is a negative number. All of the following must be positive except:

Explanation:

Since  and  are positive, all powers of  and  will be positive; also, in each of the expressions, the powers of  and  are being added. The clue to look for is the power of  and the sign before it.

In the cases of  and , since the negative number  is being raised to an even power, each expression amounts to the sum of three positive numbers, which is positive.

In the cases of  and , since the negative number  is being raised to an odd power, the middle power is negative - but since it is being subtracted, it is the same as if a positive number is being added. Therefore, each is essentially the sum of three positive numbers, which, again, is positive.

In the case of , however, since the negative number  is being raised to an odd power, the middle power is again negative. This time, it is basically the same as subtracting a positive number. As can be seen in this example, it is possible to have this be equal to a negative number:

:

Therefore,  is the correct choice.

### Example Question #25 : Integers

Let  be a negative integer and  be a nonzero integer. Which of the following must be negative regardless of whether  is positive or negative?

None of the other answers is correct.

Explanation:

Since  is positive, , the product of a negative number and a positive number, must be negative also.

Of the others:

is incorrect; if  is negative, then  is positive, and  assumes the sign of .

is incorrect; again,  is positive, and if  is a positive number,  is positive.

is incorrect; regardless of the sign of  is positive, and if its absolute value is greater than that of  is positive.

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