# GMAT Math : Absolute Value

## Example Questions

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### Example Question #1 : Dsq: Understanding Absolute Value

Given that , evaluate .

1)

2)

BOTH statements TOGETHER are 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 1 or Statement 2 ALONE is sufficient to answer the question.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is not sufficient.

Statement 1 ALONE is sufficient to answer the question, but Statement 2 ALONE is not sufficient.

Explanation:

,

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 #1 : Absolute Value

Using the following statements, Solve for

(read as  equals the absolute value of  minus )

1.

2.

Statements (1) and (2) TOGETHER are NOT sufficient.

EACH statement ALONE is sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is 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

Explanation:

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 #3 : Absolute Value

Is

(1)

(2)

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

EACH statement ALONE is sufficient.

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

Statements (1) and (2) TOGETHER are NOT sufficient.

Explanation:

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 #4 : Absolute Value

Is nonzero number  positive or negative?

Statement 1:

Statement 2:

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.

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.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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 #1 : Dsq: Understanding Absolute Value

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.

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.

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.

Explanation:

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 #1 : Dsq: Understanding 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.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER 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.

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

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 #7 : 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 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.

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.

Explanation:

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 #1 : 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.

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 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 2 ALONE is sufficient to answer the question, but Statement 1 ALONE is NOT sufficient to answer the question.

Explanation:

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 #1 : 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.

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.

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.

Explanation:

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 #10 : Absolute Value

is a real number. 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.

EITHER 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.

Statement 2 ALONE is sufficient to answer the question, but Statement 1 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.

Explanation:

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 .

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