### All GMAT Math Resources

## Example Questions

### Example Question #1 : Dsq: Solving Inequalities

Data sufficiency question- do not actually solve the question

Is ?

1.

2.

**Possible Answers:**

Statements 1 and 2 together are not sufficient, and additional information is needed to answer the question

Both statements taken together are sufficient to answer the question but neither statement alone is sufficient

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

Statement 1 alone is sufficient, but statement 2 along is not sufficient to answer the question

Each statement alone is sufficient

**Correct answer:**

Statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

From statement 1, we can conclude that but not . From the second statement, we can conclude that the greatest product will result from or 9, which is less than 12.

### Example Question #1 : Dsq: Solving Inequalities

How many solutions does the equation have?

Statement 1:

Statement 2:

**Possible Answers:**

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.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

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

If we only know that , then the above statement becomes , and it can have zero, one, or two solutions depending on the value of . For example:

If , the equation is , which has no solution, as an absolute value cannot be negative.

If , the equation is , which requires that , or , since only 0 has absolute value 0; this means the equation has one solution.

If we only know that , then the equation becomes , which has no solution regardless of the value of ; this is because, as stated before, an absolute value cannot be negative.

### Example Question #2 : Dsq: Solving Inequalities

True or false: is a positive number.

Statement 1:

Statement 2:

**Possible Answers:**

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.

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:**

If is negative, then and . Therefore, either Statement 1 or Statement 2 alone proves nonnegative. However, if , then , but is false.

Therefore, Statement 2 proves positive, but Statement 1 only proves nonnegative.

### Example Question #11 : Algebra

True or false:

Statement 1:

Statement 2:

**Possible Answers:**

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.

BOTH statements TOGETHER are insufficient to answer the question.

**Correct answer:**

BOTH statements TOGETHER are insufficient to answer the question.

makes both statements true, since and .

makes both statements true, since and .

One of the two values is less than 5, and one is greater than 5. The statements together provide insufficient information.

### Example Question #1 : Inequalities

is a whole number.

True or false: is odd.

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:**

Statement 1 alone is a superfluous statement, since a positive number raised to any power must yield a positive result.

Statement 2 alone answers the question, since a negative number raised to a whole number exponent yields a positive result if and only if the exponent is even. Since Statement 2 states that is positive, is even, not odd.

### Example Question #1 : Inequalities

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.

Assume Statement 1 alone. can be rewritten as .

Assume Statement 2 alone. It can be rewritten as

the solution set of which is

From either statement alone, it follows that .

### Example Question #811 : Data Sufficiency Questions

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:**

EITHER statement ALONE is sufficient to answer the question.

Assume Statement 1 alone. Since and are both positive, we can divide both sides by to yield the statement

Since increases as increases, and since , it follows that .

Assume Statement 2 alone. Since the cube root of a number assumes the same sign as the number itself, implies that .

From either statement alone it follows that .

### Example Question #1 : Inequalities

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:**

Assume Statement 1 only. Both and 12 make the statement true, since . But one is less than 11 and one is not.

Assume Statement 2 only. Then, since an odd (third) root of a number assumes the sign of that number, and an odd root function is an increasing function, we can simply take the cube root of each side:

or

.

### Example Question #2 : Inequalities

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:**

EITHER statement ALONE is sufficient to answer the question.

Assume Statement 1 alone. Since the fifth (odd) power of a number assumes the same sign as the number itself, and have the same sign, and implies that .

Assume Statement 2 alone. Since and are both positive, we can divide both sides by to yield the statement

Since increases as does, and since , it follows that .

### Example Question #2 : Dsq: Solving Inequalities

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:**

BOTH statements TOGETHER are insufficient to answer the question.

Both statements together provide insufficient information. For example,

If , then:

If , then

Both values fit the conditions of both statements, but only one is greater than . The question is not answered.

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