GMAT Math : Inequalities

Example Questions

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Example Question #1 : Inequalities

Data sufficiency question- do not actually solve the question

Is ?

1.

2.

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

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

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

Each statement alone is sufficient

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

Explanation:

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 #14 : Algebra

How many solutions does the equation  have?

Statement 1:

Statement 2:

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.

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

Explanation:

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 #11 : Algebra

True or false:  is a positive number.

Statement 1:

Statement 2:

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.

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.

Explanation:

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 #1 : Dsq: Solving Inequalities

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.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

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 : Dsq: Solving Inequalities

is a whole number.

True or false:  is odd.

Statement 1:

Statement 2:

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.

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.

Explanation:

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 #18 : Algebra

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.

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.

EITHER statement ALONE is sufficient to answer the question.

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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 #1 : Dsq: Solving Inequalities

True or false:

Statement 1:

Statement 2:

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.

EITHER 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.

Explanation:

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 : Dsq: Solving Inequalities

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.

Explanation:

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:

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.

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.

Explanation:

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 #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.

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 sufficient to answer the question, but NEITHER statement ALONE is sufficient to answer the question.

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

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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