GMAT Math : Algebra

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Example Questions

Example Question #11 : Algebra

is a number not in the set $\left\{-1, 0, 1 \right\}$ .

Of the elements $\left \{ x^{2}, x^{3}, x^{4}\right \}$ , which is the greatest?

Statement 1: is a negative number.

Statement 2: $\left | x\right |>\frac{1}{2}$

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 insufficient 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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Assume both statements are known. The greatest of the three numbers must be $x^{2}$ or $x^{4}$ , since even powers of negative numbers are positive and odd powers of negative numbers are negative.

Case 1: $x = -\frac{2}{3}$

$x^{2} = \left ( -\frac{2}{3} \right )^{2} = \frac{4}{9} = \frac{36}{81}$

$x^{4} = \left ( -\frac{2}{3} \right )^{4} = \frac{16}{81}$

$x^{2} > x^{4}> x^{3}$

Case 2: x = -2 ,

then

$x^{2} = \left ( -2 \right )^{2} =4$

$x^{4} = \left (-2 \right )^{4} = 16$

$x^{4} > x^{2} > x^{3}$

In both cases, is negative and $\left | x\right |>\frac{1}{2}$ , but in one case, $x^{2}$ is the greatest number, and in the other, $x^{4}$ is. The two statements together are inconclusive.

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Example Question #12 : Algebra

Philip has been assigned to write one number in the circle and one number in the square in the diagram below in order to produce a number in scientifc notation.

$\bigcirc \times 10^{\square }$ .

Did Philip succeed?

Statement 1: Philip wrote -7.5 in the circle.

Statement 2: Philip wrote in the square.

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.

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:

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

Explanation:

A number in scientific notation takes the form

$a \times 10 ^{n}$

where $1 \le |a| < 10$ and is an integer of any sign.

Statement 1 alone proves that Philip entered a correct number into the circle, since $1 \le |-7.5| = 7.5 < 10$ . Statement 2 alone proves that he entered a correct number into the square, since is an integer. But each statement alone is insufficient, since each leaves uinclear whether the other number is valid. The two statements together, however, prove that Philip put correct numbers in both places, thereby writing a number in scientific notation.

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Example Question #801 : Data Sufficiency Questions

is an integer. Is there a real number such that $x^{A}=B$ ?

Statement 1: is negative

Statement 2: is even

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.

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.

Correct answer:

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

Explanation:

The equivalent question is "does have a real $A \textrm{th }$ root?"

If you know only that is negative, you need to know whether is even or odd; negative numbers have real odd-numbered roots, but not real even-numbered roots.

If you know only that is even, you need to know whether is negative or nonnegative; negative numbers do not have real even-numbered roots, but nonnegative numbers do.

If you know both, however, then you know that the answer is no, since as stated before, negative numbers do not have real even-numbered roots.

Therefore, the answer is that both statements together are sufficient to answer the question, but neither statement alone is sufficient to answer the question.

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Example Question #11 : Algebra

Data sufficiency question- do not actually solve the question

Is $\small xy< 12$ ?

1. $\small x^2=64; 1\leq y\leq 1.5$

2. $\small x+y=6$

Possible Answers:

Each statement alone is sufficient

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 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

Correct answer:

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

Explanation:

From statement 1, we can conclude that $\small xy\leq 12$ but not $\small xy< 12$ . From the second statement, we can conclude that the greatest product will result from $\small 3+3=6$ or 9, which is less than 12.

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Example Question #12 : Algebra

How many solutions does the equation $\left | x - A \right | = B$ have?

Statement 1: A = 8

Statement 2: B = -8

Possible Answers:

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.

EITHER statement ALONE is sufficient 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.

Explanation:

If we only know that A = 8 , then the above statement becomes $\left | x - 8 \right | = B$ , and it can have zero, one, or two solutions depending on the value of . For example:

If B = -5 , the equation is $\left | x - 8 \right | = -5$ , which has no solution, as an absolute value cannot be negative.

If B = 0 , the equation is $\left | x - 8 \right | = 0$ , which requires that x - 8=0 , or x = 8 , since only 0 has absolute value 0; this means the equation has one solution.

If we only know that B = -8 , then the equation becomes $\left | x - A \right | = -8$ , which has no solution regardless of the value of ; this is because, as stated before, an absolute value cannot be negative.

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Example Question #13 : Algebra

True or false: is a positive number.

Statement 1: $N = \left | N \right |$

Statement 2: $N > N^{2}$

Possible Answers:

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.

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.

Correct answer:

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 |N| = -N and $N ^{2} > 0 > N$ . Therefore, either Statement 1 or Statement 2 alone proves nonnegative. However, if N = 0 , then $N = \left | N \right |$ , but $N > N^{2}$ is false.

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

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Example Question #14 : Algebra

True or false: x < 5

Statement 1: |x| > 5

Statement 2: $x^{2}> 25$

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 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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

x= 6 makes both statements true, since |6|= 6 > 5 and $6^{2}= 36 > 25$ .

x= - 6 makes both statements true, since |-6|= 6 > 5 and $(-6)^{2}= 36 > 25$ .

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

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Example Question #15 : Algebra

is a whole number.

True or false: is odd.

Statement 1: $2^{x} > 0$

Statement 2: $(-2)^{x} > 0$

Possible Answers:

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.

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.

Correct answer:

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 $(-2)^{x}$ is positive, is even, not odd.

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Example Question #16 : Algebra

True or false: x < 9

Statement 1: |x| < 9

Statement 2: $x^{2}< 81$

Possible Answers:

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Assume Statement 1 alone. |x| < 9 can be rewritten as -9 < x < 9 .

Assume Statement 2 alone. It can be rewritten as

$x^{2}< 81$

the solution set of which is -9 < x < 9

From either statement alone, it follows that x < 9 .

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Example Question #17 : Algebra

True or false: x > 0

Statement 1: $2^{x}<4^{x}$

Statement 2: $x^{3}> 0$

Possible Answers:

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 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 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.

Explanation:

Assume Statement 1 alone. Since $2^{x}$ and $4^{x}$ are both positive, we can divide both sides by $2^{x}$ to yield the statement

$2^{x}<4^{x}$

$4^{x} > 2^{x}$

$\frac{ 4^{x}}{2^{x}} >\frac{ 2^{x}}{2^{x}}$

$\left (\frac{ 4}{2} \right )^{x} >1$

$2^{x}> 1$

Since $f(x) = 2^{x}$ increases as increases, and since $2^{0}= 1$ , it follows that x > 0 .

Assume Statement 2 alone. Since the cube root of a number assumes the same sign as the number itself, $x^{3}> 0$ implies that x>0 .

From either statement alone it follows that x>0 .

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #11 : Algebra

Example Question #12 : Algebra

Example Question #801 : Data Sufficiency Questions

Example Question #11 : Algebra

Example Question #12 : Algebra

Example Question #13 : Algebra

Example Question #14 : Algebra

Example Question #15 : Algebra

Example Question #16 : Algebra

Example Question #17 : Algebra

All GMAT Math Resources

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