DSQ: Understanding absolute value - GMAT Quantitative
Card 0 of 120
Of distinct integers 
, which is the greatest of the three?
Statement 1: 
Statement 2: 
and 
are negative.
Of distinct integers
Statement 1:
Statement 2:
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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. 
, and from Statement 1, 
; by transitivity, 
. From Statement 2, 
. This makes 
the greatest of the three.
Statement 1 alone gives insufficient information.
Case 1:
Case 2:
But in the first case,
Statement 2 gives insuffcient information, since no information is given about the sign of
Assume both statements to be true.
Is 
(1) 
(2) 
Is
(1)
(2)
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For statement (1), since we don’t know the value of 
and 
, we have no idea about the value of 
and 
.
For statement (2), since we don’t know the sign of 
and 
, we cannot compare 
and 
.
Putting the two statements together, if 
and 
, then 
.
But if 
and 
, then 
.
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.
For statement (1), since we don’t know the value of
For statement (2), since we don’t know the sign of
Putting the two statements together, if
But if
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.
Given that 
, evaluate 
.
-
-
Given that
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,
so, if we know 
and 
, then the above becomes
and 
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.
so, if we know
and
If we know
The answer is that Statement 1 alone is sufficient, but not Statement 2.
Using the following statements, Solve for 
.
(read as 
equals the absolute value of 
minus 
)
1. 
2. 
Using the following statements, Solve for
1.
2.
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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 
.
This question tests your understanding of absolute value. You should know that
Therefore, statement 1 alone is enough to solve for
Is nonzero number 
positive or negative?
Statement 1: 
Statement 2: 
Is nonzero number
Statement 1:
Statement 2:
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If we assume that 
, then it follows that:
Since we know 
, we know 
is positive, and 
and 
are negative.
If we assume that 
, then it follows that:
Since we know 
, we know 
is positive. 
is also positive and 
is negative; since 
is less than a negative number, 
is also negative.
If we assume that
Since we know
If we assume that
Since we know
is a real number. True or false: 
Statement 1: 
Statement 2: 
Statement 1:
Statement 2:
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Statement 1 and Statement 2 are actually equivalent.
If 
, then either 
or 
by definition.
If 
, then either 
or 
.
The correct answer is that the two statements together are not enough to answer the question.
Statement 1 and Statement 2 are actually equivalent.
If
If
The correct answer is that the two statements together are not enough to answer the question.
is a real number. True or false: 
Statement 1: 
Statement 2: 
Statement 1:
Statement 2:
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If 
, then we can deduce only that either 
or 
. Statement 1 alone does not answer the question.
If 
, 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 
.
If
If
True or false: 
Statement 1: 
Statement 2: 
True or false:
Statement 1:
Statement 2:
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Statement 1 and Statement 2 are actually equivalent.
If 
, then either 
by definition.
If 
, then either 
.
From either statement alone, it can be deduced that 
.
Statement 1 and Statement 2 are actually equivalent.
If
If
From either statement alone, it can be deduced that
is a real number. True or false: 
Statement 1: 
Statement 2: 
Statement 1:
Statement 2:
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If 
, then, by definition, 
.
If Statement 1 is true, then
,
so 
must be in the desired range.
If Statement 2 is true, then
and 
is not necessarily in the desired range.
If
If Statement 1 is true, then
so
If Statement 2 is true, then
and
is a real number. True or false: 
Statement 1: 
Statement 2: 
Statement 1:
Statement 2:
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If 
, then, by definition, 
.
If Statement 1 holds, that is, if 
, one of two things happens:
If 
is positive, then 
.
If 
is negative, then 
.
is a false statement.
If Statement 2 holds, that is, if 
, we know that 
is positive, and
is a false statement.
If
If Statement 1 holds, that is, if
If
If
If Statement 2 holds, that is, if
is a real number. True or false: 
Statement 1: 
Statement 2: 
Statement 1:
Statement 2:
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If 
, 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 .
If
If Statement 1 is true, then
Statement 1 alone does not answer the question, as
If Statement 2 is true, then
Statement 2 alone does not answer the question, as
If both statements are true, then
Of distinct integers 
, which is the greatest of the three?
Statement 1: 
Statement 2: 
Of distinct integers
Statement 1:
Statement 2:
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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.
The two statements together are insufficient.
For example, let
Therefore, either
In either case, Statement 2 is shown to be true, since
and
But if
Which, if either, is the greater number: 
or 
?
Statement 1: 
Statement 2: 
Which, if either, is the greater number:
Statement 1:
Statement 2:
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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.
Statement 1 alone gives insufficient information, as is seen in these two cases:. For example, if
However, if
Therefore, it is not clear which, if either, of
Now assume Statement 2 alone.
If
Therefore,
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.
Let
At how many points does the graph of the function
intersect the
Statement 1:
Statement 2:
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To determine the point(s), if any, at which the graph of a function 
intersects the 
-axis, set 
and solve for 
.
At this point, we can examine the equation. Since the absolute value of a number must be nonnegative, the sign of 
tells us how many solutions exist to this equation. If 
, there is no solution, and therefore, the graph of 
does not intersect the 
-axis. If 
, then there is one solution, and, therefore, the graph of 
intersects the 
-axis at exactly one point. If 
, then there are two solutions, and, therefore, the graph of 
intersects the 
-axis at exactly two points.
To determine the sign of 
, 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 
and 
, so the graph of 
can be determined to not cross the 
-axis at all.
To determine the point(s), if any, at which the graph of a function
At this point, we can examine the equation. Since the absolute value of a number must be nonnegative, the sign of
To determine the sign of
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: 
Let
At how many points does the graph of the function
intersect the
Statement 1:
Statement 2:
Tap to see back →
To determine the point(s), if any, at which the graph of a function 
intersects the 
-axis, set 
and solve for 
.
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 
. 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.
To determine the point(s), if any, at which the graph of a function
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
Of distinct integers , which is the greatest of the three?
Statement 1:
Statement 2: and are negative.
Of distinct integers
Statement 1:
Statement 2:
Tap to see back →
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. , and from Statement 1, ; by transitivity, . From Statement 2, . This makes the greatest of the three.
Statement 1 alone gives insufficient information.
Case 1:
Case 2:
But in the first case,
Statement 2 gives insuffcient information, since no information is given about the sign of
Assume both statements to be true.
Is
(1)
(2)
Is
(1)
(2)
Tap to see back →
For statement (1), since we don’t know the value of and , we have no idea about the value of and .
For statement (2), since we don’t know the sign of and , we cannot compare and .
Putting the two statements together, if and , then .
But if and , then .
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.
For statement (1), since we don’t know the value of
For statement (2), since we don’t know the sign of
Putting the two statements together, if
But if
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.
Given that , evaluate .
-
-
Given that
Tap to see back →
,
so, if we know and , then the above becomes
and
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.
so, if we know
and
If we know
The answer is that Statement 1 alone is sufficient, but not Statement 2.
Using the following statements, Solve for .
(read as equals the absolute value of minus )
1.
2.
Using the following statements, Solve for
1.
2.
Tap to see back →
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 .
This question tests your understanding of absolute value. You should know that
Therefore, statement 1 alone is enough to solve for
Is nonzero number positive or negative?
Statement 1:
Statement 2:
Is nonzero number
Statement 1:
Statement 2:
Tap to see back →
If we assume that , then it follows that:
Since we know , we know is positive, and and are negative.
If we assume that , then it follows that:
Since we know , we know is positive. is also positive and is negative; since is less than a negative number, is also negative.
If we assume that
Since we know
If we assume that
Since we know