### All PSAT Math Resources

## Example Questions

### 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 #1 : Negative Numbers

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 and III only

I, II and III

**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.

### Example Question #26 : Integers

Given that are both integers, , and , which of the following is correct about the sign of the expression ?

**Possible Answers:**

The expression must be negative.

The expression can be positive, negative, or zero.

The expression must be negative or zero.

The expression must be positive.

The expression must be positive or zero.

**Correct answer:**

The expression must be negative or zero.

If , then we know that is any number between or equal to and . Therefore must be a negative number.

Also, if , then we know that is any number between or equal to and . Therefore must be a negative number.

Now looking the expression we can find the sign of each component in the expression.

Since is negative, we know that a negative number minus another number is still a negative number.

Therefore, is a negative number.

Since ** **is between or equal to and we can plug in these end values in to determine the sign of .

Therefore, is either zero or a positive number.

Now to find the sign of the expression we look at the product of the two components. The product of a negative number and a positive number is a negative number; the product of a negative number and zero is zero. Therefore, the correct choice is that is negative or zero.

### Example Question #21 : Integers

Find the product.

**Possible Answers:**

**Correct answer:**

When multiplying together two negatives, our value for the product become positive.

### Example Question #28 : Integers

Find the product.

**Possible Answers:**

**Correct answer:**

Since we have one positive and one negative multiple, the resulting product must be negative.

### Example Question #3 : Negative Numbers

If x is a negative integer, what else must be a negative integer?

**Possible Answers:**

x²

x – x

x – (–x)

x² – x

**Correct answer:**

x – (–x)

By choosing a random negative number, for example: –4, we can input the number into each choice and see if we come out with another negative number. When we put –4 in for x, we would have –4 – (–(–4)) or –4 – 4, which is –8. Plugging in the other options gives a positive answer. You can try other negative numbers, if needed, to confirm this still works.

### Example Question #21 : Integers

–7 – 7= x

–7 – (–7) = y

what are x and y, respectively

**Possible Answers:**

x = –14, y = 14

x = –14, y = 0

y = 0, x = 14

x = 0, y = 0

x = 14, y = –14

**Correct answer:**

x = –14, y = 0

x: –7 – 7= –7 + –7 = –14

y: –7 – (–7) = –7 + 7 = 0

when subtracting a negative number, turn it into an addition problem

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