All GMAT Math Resources
Example Questions
Example Question #1 : Dsq: Understanding Absolute Value
Given that , evaluate .
1)
2)
EITHER Statement 1 or Statement 2 ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient.
BOTH statements TOGETHER are NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient.
,
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.
Example Question #2 : Dsq: Understanding Absolute Value
Using the following statements, Solve for .
(read as equals the absolute value of minus )
1.
2.
EACH statement ALONE is sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient
Statements (1) and (2) TOGETHER are NOT sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient
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 .
Example Question #2 : Dsq: Understanding Absolute Value
Is
(1)
(2)
Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
EACH statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
Statements (1) and (2) TOGETHER are NOT sufficient.
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.
Example Question #3 : Dsq: Understanding Absolute Value
Is nonzero number positive or negative?
Statement 1:
Statement 2:
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
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.
Example Question #4 : Dsq: Understanding Absolute Value
True or false:
Statement 1:
Statement 2:
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
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 .
Example Question #6 : Absolute Value
is a real number. True or false:
Statement 1:
Statement 2:
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
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.
Example Question #5 : Dsq: Understanding Absolute Value
is a real number. True or false:
Statement 1:
Statement 2:
BOTH statements TOGETHER are insufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
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.
Example Question #6 : Dsq: Understanding Absolute Value
is a real number. True or false:
Statement 1:
Statement 2:
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
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 .
Example Question #3 : Dsq: Understanding Absolute Value
is a real number. True or false:
Statement 1:
Statement 2:
BOTH statements TOGETHER are insufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
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.
Example Question #8 : Absolute Value
is a real number. True or false:
Statement 1:
Statement 2:
Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is NOT sufficient to answer the question.
Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.
EITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are insufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
BOTH statements TOGETHER are sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.
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 .