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## Example Questions

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

Given that , evaluate .

1)

2)

**Possible Answers:**

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

BOTH statements TOGETHER are NOT 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.

**Correct answer:**

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.

**Possible Answers:**

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.

**Correct answer:**

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 #3 : Dsq: Understanding Absolute Value

Is

(1)

(2)

**Possible Answers:**

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

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

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

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

EACH statement ALONE is sufficient.

**Correct answer:**

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 #4 : Dsq: Understanding Absolute Value

Is nonzero number positive or negative?

Statement 1:

Statement 2:

**Possible Answers:**

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 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.

**Correct answer:**

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 #5 : Dsq: Understanding Absolute Value

True or false:

Statement 1:

Statement 2:

**Possible Answers:**

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.

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.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

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 : Dsq: Understanding Absolute Value

is a real number. True or false:

Statement 1:

Statement 2:

**Possible Answers:**

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.

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.

**Correct answer:**

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 #7 : Dsq: Understanding Absolute Value

is a real number. True or false:

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

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 #8 : Dsq: Understanding Absolute Value

is a real number. True or false:

Statement 1:

Statement 2:

**Possible Answers:**

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

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 #9 : Dsq: Understanding Absolute Value

is a real number. True or false:

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

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 #10 : Dsq: Understanding Absolute Value

is a real number. True or false:

Statement 1:

Statement 2:

**Possible Answers:**

BOTH statements TOGETHER are insufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

**Correct answer:**

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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