# GMAT Math : DSQ: Solving inequalities

## Example Questions

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### Example Question #21 : Algebra

The variables  stand for integer quantities.

Order from least to greatest:

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.

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.

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

Explanation:

Statement 1 only gives us the order of the bases; we cannot order the powers without any information about the exponents. Similarly, Statement 2 alone only tells us the common exponent; without knowing the order of the bases, we cannot order the powers.

Assume both statements are true. Let's look at  and .

Since  and  are both positive, and , we can apply the multiplication property of equality:

Similarly,  etc.

So, if  and ,

Since  from Statement 1, it follows that

which, from Statement 2, can be rewritten as

.

The order has been determined.

### Example Question #21 : Algebra

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.

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 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:

Assume Statement 1 alone. Then, since an odd (fifth) root of a number assumes the sign of that number, and an odd root function is an increasing function, we can simply take the fifth root of each side:

,

Assume Statement 2 alone.By a similar argument,

,

From either statement alone, it follows that  is true.

### Example Question #23 : Algebra

The variables  stand for integer quantities.

Order from least to greatest:

Statement 1:

Statement 2:

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.

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.

Explanation:

We show the staements together provide insufficient information by examining two situations.

Suppose .

If , the four expressions become:

and

If , the four expressions become:

In ascending order, the expressions are .

Since the orderings are different in the two cases, the two statements together give insufficient information as to their correct ordering.

### Example Question #22 : Algebra

The variables  stand for integer quantities.

Order from least to greatest:

Statement 1:

Statement 2:

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.

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

Explanation:

Statement 1 alone only tells us the common base; without knowing the order of the exponents, we cannot order the powers. Similarly, Statement 2 only gives us the order of the exponents; we cannot order the powers without any information about the bases.

Assume both statements are true. Since , it holds that

et cetera, and

and so forth in both directions. That is,

Therefore, since , it follows that

,

and the ordering is determined.

### Example Question #823 : Data Sufficiency Questions

True or false:

Statement 1:

Statement 2:  is positive.

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.

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.

Explanation:

Assume Statement 1 alone. From , it can be determined that either  or , but nothing more; the statement alone provides insufficient information. Statement 2 taken alone provides insufficient information as well, since the positive numbers include numbers both less than and greater than 7.

Assume both statements are true. From Statement 1, either  or , but Statement 2 gives us that . Therefore, , and the question is answered.

### Example Question #24 : Algebra

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.

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

Explanation:

The absolute value of a number is its unsigned value - that is, if the number is nonnegative, it is the number itself, and if the number is negative, it is the corresponding positive.

From Statement 1 alone, since , then it follows that  - this is equivalent to saying .

Statement 2 alone provides insufficient information. For example:

If , then

if , then

Both numbers fit the condition of Statement 2, but

and

### Example Question #11 : Dsq: Solving Inequalities

The variables  stand for integer quantities.

Order from least to greatest:

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.

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.

EITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

We show the two statements provide insufficient information by assuming both statements to be true and showing that the ordering is different depending on the common exponent. In both cases, we let .

If , then the expressions become

The correct ordering is .

If , then the expressions become

The correct ordering is .

### Example Question #11 : Dsq: Solving Inequalities

is a real number. True or false:  is positive.

Statement 1:

Statement 2:

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. If  is positive, then  and ; since  is the sum of three positive numbers, then , and  is a false statement. Therefore,  cannot be positive.

Assume Statement 2. If  is positive, then so is , and the inequality can be rewritten as

Consequently,

,

a contradiction since  is positive. Therefore,  is not positive.

### Example Question #32 : Algebra

is a real number. True or false:  is positive.

Statement 1:

Statement 2:

BOTH STATEMENTS TOGETHER do NOT provide sufficient information to answer the question.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER provide sufficient information to answer the question, but NEITHER STATEMENT ALONE provides sufficient information to answer the question.

STATEMENT 2 ALONE provides sufficient information to answer the question, but STATEMENT 1 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

STATEMENT 1 ALONE provides sufficient information to answer the question, but STATEMENT 2 ALONE does NOT provide sufficient information to answer the question.

Explanation:

Assume Statement 1 alone. , or . For any base , if , then , so it follows that  is therefore positive.

Assume Statement 2 alone. Both  and  are solutions:

and

The sign of  cannot be determined.

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