### All PSAT Math Resources

## Example Questions

### Example Question #12 : Integers

How many elements of the set are less than ?

**Possible Answers:**

One

Four

Three

Two

None

**Correct answer:**

Four

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?

**Possible Answers:**

b > 0

ac < 0

a + b < 0

a – b > 0

a > 0

**Correct answer:**

a + b < 0

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 ?

**Possible Answers:**

45

**Correct 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.

### Example Question #1 : Negative Numbers

Solve for :

**Possible Answers:**

**Correct answer:**

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

### Example Question #2 : Negative Numbers

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

**Possible Answers:**

**Correct answer:**

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)

**Possible Answers:**

I only

All of these

None of these

II only

III only

**Correct answer:**

All of these

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 #22 : Integers

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

I)

II)

III)

**Possible Answers:**

None of I, II or III

I and II only

II and III only

I, II and III

I and III only

**Correct answer:**

None of I, II or III

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?

**Possible Answers:**

**Correct answer:**

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:

**Possible Answers:**

**Correct answer:**

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?

**Possible Answers:**

None of the other answers is correct.

**Correct answer:**

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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