DSQ: Understanding absolute value - GMAT Quantitative Reasoning

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Question

Is $\left | a-b \right |<\left | a-c \right |?$

(1)

(2) $\left | b \right |<\left | c \right |$

Answer

For statement (1), since we don’t know the value of and , we have no idea about the value of $\left | a-b \right |$ and $\left | a-c \right |$ .

For statement (2), since we don’t know the sign of and , we cannot compare $\left | a-b \right |$ and $\left | a-c \right |$ .

Putting the two statements together, if and , then $\left | a-b \right |<\left | a-c \right |$ .

But if and , then $\left | a-b \right |>\left | a-c \right |$ .

Therefore, we cannot get the only correct answer for the questions, suggesting that the two statements together are not sufficient. For this problem, we can also plug in actual numbers to check the answer.

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Question

Given that , evaluate .

$x^{2} - y^{2} = 40$

Answer

$x^{2} - y^{2} = (x + y) (x - y)$ ,

so, if we know and $x^{2} - y^{2} = 40$ , then the above becomes

and $x - y = 40 \div 10 = 4$

If we know and , then we need two numbers whose sum is 10 and whose product is 21; by inspection, these are 3 and 7. However, we do not know whether and or vice versa just by knowing their sum and product. Therefore, either , or .

The answer is that Statement 1 alone is sufficient, but not Statement 2.

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Question

Using the following statements, Solve for .

(read as equals the absolute value of minus )

1.

2.

Answer

This question tests your understanding of absolute value. You should know that

since we are finding the absolute value of the difference. We can prove this easily. Since , we know their absolute values have to be the same.

Therefore, statement 1 alone is enough to solve for . and we get .

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Question

Is nonzero number positive or negative?

Statement 1:

Statement 2: $N^{2} + 5N < 0$

Answer

If we assume that , then it follows that:

Since we know , we know is positive, and and are negative.

If we assume that $N^{2} + 5N < 0$ , then it follows that:

$N^{2} + 5N -N^{2} < 0-N^{2}$

$5N < -N^{2}$

$N < -\frac{N^{2}}{5}$

Since we know , we know $N^{2}$ is positive. $\frac{N^{2}}{5}$ is also positive and $-\frac{N^{2}}{5}$ is negative; since is less than a negative number, is also negative.

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Question

is a real number. True or false:

Statement 1: $\left | N\right | > 10$

Statement 2: $N^{2} > 100$

Answer

Statement 1 and Statement 2 are actually equivalent.

If $\left | N\right | > 10$ , then either or by definition.

If $N^{2} > 100$ , then either or .

The correct answer is that the two statements together are not enough to answer the question.

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Question

is a real number. True or false:

Statement 1: $\left | N\right | > 5$

Statement 2: $N^{3} > 125$

Answer

If $\left | N\right | > 5$ , then we can deduce only that either or . Statement 1 alone does not answer the question.

If $N^{3} > 125$ , then must be positive, as no negative number can have a positive cube. The positive numbers whose cubes are greater than 125 are those greater than 5. Therefore, Statement 2 alone proves that .

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Question

True or false:

Statement 1: $\left | N\right | <7$

Statement 2: $N^{2} < 49$

Answer

Statement 1 and Statement 2 are actually equivalent.

If $\left | N\right | <7$ , then either by definition.

If $N^{2} < 49$ , then either .

From either statement alone, it can be deduced that .

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Question

is a real number. True or false: $\left | N\right | < 5$

Statement 1: $\left | N - 1\right | < 4$

Statement 2: $\left | N +1 \right | <6$

Answer

If $\left | N\right | < 5$ , then, by definition, .

If Statement 1 is true, then

$\left | N - 1\right | < 4$

,

so must be in the desired range.

If Statement 2 is true, then

$\left | N +1 \right | <6$

and is not necessarily in the desired range.

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Question

is a real number. True or false: $\left | N\right | < 6$

Statement 1: $N^{2} > 36$

Statement 2: $N^{3} > 216$

Answer

If $\left | N\right | < 6$ , then, by definition, .

If Statement 1 holds, that is, if $N^{2} > 36$ , one of two things happens:

If is positive, then $N > \sqrt{36} = 6$ .

If is negative, then $N < - \sqrt{36} = -6$ .

$\left | N\right | < 6$ is a false statement.

If Statement 2 holds, that is, if $N^{3} > 216$ , we know that is positive, and

$N >\sqrt[3]{ 216} = 6$

$\left | N\right | < 6$ is a false statement.

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Question

is a real number. True or false: $\left | N \right | < 5$

Statement 1:

Statement 2:

Answer

If $\left | N\right | < 5$ , then, by definition, - that is, both and .

If Statement 1 is true, then

Statement 1 alone does not answer the question, as follows, but not necessarily .

If Statement 2 is true, then

Statement 2 alone does not answer the question, as follows, but not necessarily .

If both statements are true, then and both follow, and , meaning that $\left | N \right | < 5$ .

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Question

Of distinct integers , which is the greatest of the three?

Statement 1:

Statement 2:

Answer

The two statements together are insufficient.

For example, let . Then, from Statement 2,

Therefore, either or .

In either case, Statement 2 is shown to be true, since

and

But if , then is the greatest of the three. If , then is the greatest. Therefore,the two statements together are not enough.

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Question

Of distinct integers , which is the greatest of the three?

Statement 1:

Statement 2: and are negative.

Answer

Statement 1 alone gives insufficient information.

Case 1:

, which is true.

Case 2:

, which is true.

But in the first case, is the greatest of the three. In the second, is the greatest.

Statement 2 gives insuffcient information, since no information is given about the sign of .

Assume both statements to be true. $|a| \ge 0$ , and from Statement 1, ; by transitivity, . From Statement 2, . This makes the greatest of the three.

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Question

Which, if either, is the greater number: or ?

Statement 1:

Statement 2:

Answer

Statement 1 alone gives insufficient information, as is seen in these two cases:. For example, if , then

However, if , then

Therefore, it is not clear which, if either, of and is greater.

Now assume Statement 2 alone.

If is negative, then , which, being an absolute value of a number, must be nonnegative, is the greater number. If is positive, then so is , so

.

Therefore,

.

is the greater number in either case.

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Question

Let be any three (not necessarily distinct) integers.

At how many points does the graph of the function

intersect the -axis?

Statement 1: and are nonzero integers of opposite sign.

Statement 2: and are nonzero integers of opposite sign.

Answer

To determine the point(s), if any, at which the graph of a function intersects the -axis, set and solve for .

$|x-B | = - \frac{C}{A}$

At this point, we can examine the equation. Since the absolute value of a number must be nonnegative, the sign of $- \frac{C}{A}$ tells us how many solutions exist to this equation. If $- \frac{C}{A} < 0$ , there is no solution, and therefore, the graph of does not intersect the -axis. If $- \frac{C}{A} = 0$ , then there is one solution, and, therefore, the graph of intersects the -axis at exactly one point. If $- \frac{C}{A} > 0$ , then there are two solutions, and, therefore, the graph of intersects the -axis at exactly two points.

To determine the sign of $- \frac{C}{A}$ , we need to whether the signs of both and are like or unlike, or that . Either statement alone eliminates the possibility that , but neither alone gives the signs of both and . However, if both statements are assumed, then, since and have the opposite sign as , they have the same sign. This makes $\frac{C}{A} > 0$ and $- \frac{C}{A} < 0$ , so the graph of can be determined to not cross the -axis at all.

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Question

Let be any three (not necessarily distinct) integers.

At how many points does the graph of the function

intersect the -axis?

Statement 1:

Statement 2:

Answer

To determine the point(s), if any, at which the graph of a function intersects the -axis, set and solve for .

$|x-B | = - \frac{C}{A}$

At this point, we can examine the equation. For a solution to exist, since the absolute value of a number must be nonnegative, it must hold that $- \frac{C}{A} \geq 0$ . This happens if and are of opposite sign, or if . However, Statement 2 tells us that , and neither statement tells us the sign of . The two statements together provide insufficient information.

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