Negative Numbers - PSAT Math

Card 0 of 16

Question

a, b, c are integers.

abc < 0

ab > 0

bc > 0

Which of the following must be true?

Answer

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.

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Question

How many elements of the set $\left { -4, -3, -2, -1 \right }$ are less than $\left | - 2 \frac{1}{2} \right |$ ?

Answer

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

$\left | - 2 \frac{1}{2} \right | = 2 \frac{1}{2}$

All four (negative) numbers in the set $\left { -4, -3, -2, -1 \right }$ are less than this positive number.

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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

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

Answer

Because $\dpi{100} \small -3b$ is positive, $\dpi{100} \small b$ must be negative since the product of two negative numbers is positive.

Because $\dpi{100} \small ab$ is also positive, $\dpi{100} \small a$ must also be negative in order to produce a prositive product.

To check you answer, you can try plugging in any negative number for $\dpi{100} \small a$ .

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Question

Find the product.

$(-270) \times (-131)$

Answer

When multiplying together two negatives, our value for the product become positive. $(-270) \times (-131) = 270 \times 131 = 35370$

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Question

Find the product.

$76 \times -31=$

Answer

Since we have one positive and one negative multiple, the resulting product must be negative. $76 \times (-31) = -(76\times 31) = -2356$

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Question

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

Answer

Since $b^{2}$ is positive, $ab^{2}$ , the product of a negative number and a positive number, must be negative also.

Of the others:

$a^{2}b$ is incorrect; if is negative, then $a^{2}$ is positive, and $a^{2}b$ assumes the sign of .

$a^{2}+ b$ is incorrect; again, $a^{2}$ is positive, and if is a positive number, $a^{2}+ b$ is positive.

$a+b^{2}$ is incorrect; regardless of the sign of , $b^{2}$ is positive, and if its absolute value is greater than that of , $a+b^{2}$ is positive.

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Question

Given that are both integers, $-10 \leq a \leq -1$ , and $-10 \leq b \leq -1$ , which of the following is correct about the sign of the expression ?

Answer

If $-10 \leq a \leq -1$ , then we know that is any number between or equal to and . Therefore must be a negative number.

Also, if $-10 \leq b \leq -1$ , 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.

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Question

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

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 $a^{3}+ b^{4} +c$ and $a+ b^{2} +c^{6}$ , 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 $a^{2} -b^{7} +c$ and $a^{2}-b^{3} +c$ , 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 $a^{4}+ b^{5} +c^{3}$ , 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:

:

$a^{4}+ b^{5} +c^{3} = 1^{4}+ (-2)^{5} +1^{3} = 1 + (-32)+ 1 = -30$

Therefore, $a^{4}+ b^{5} +c^{3}$ is the correct choice.

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Question

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

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.

$\frac{ a^{6}b^{4}}{c^{9}}$ : $a^{6}$ and $b^{4}$ are positive, yielding a positive dividend; $c^{9}$ is a negative divisor; this result is negative.

$\frac{ a^{7}c^{5}}{b^{3}}$ : $a^{7}$ and $c^{5}$ are negative, yielding a positive dividend; $b^{3}$ is a negative divisor; this result is negative.

$\frac{ a^{8}c^{3}}{b^{6}}$ : $a^{8}$ is positive and $c^{3}$ is negative, yielding a negative dividend; $b^{3}$ is a positive divisor; this result is negative.

$\frac{ b^{5}c^{4}}{a^{12}}$ : $b^{5}$ is negative and $c^{4}$ is positive, yielding a negative dividend; $a^{12}$ is a positive divisor; this result is negative.

$\frac{a^{4}b^{7}}{c^{5}}$ : $a^{4}$ is positive and $b^{7}$ is negative, yielding a negative dividend; $c^{5}$ is a negative divisor; this result is positive.

The correct choice is $\frac{a^{4}b^{7}}{c^{5}}$ .

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Question

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

I) $b^{a+c}$

II) $b^{a-c}$

III) $b^{c-a}$

Answer

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.

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Question

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

I) $a^{b+c}$

II) $a^{b-c}$

III) $a^{c-b}$

Answer

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.

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Question

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

Answer

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.

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Question

–7 – 7= x

–7 – (–7) = y

what are x and y, respectively

Answer

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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Question

Answer

Subtracting a negative number is just like adding its absolute value.

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