GMAT Math : Inequalities

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Example Questions

Example Question #1 : Dsq: Solving Inequalities

Data sufficiency question- do not actually solve the question

Is $\small xy< 12$ $\displaystyle \small xy< 12$?

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

2. $\small x+y=6$ $\displaystyle \small x+y=6$

Possible Answers:

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

Each statement alone is sufficient

Statements 1 and 2 together are not sufficient, and additional information is needed 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 $\small xy\leq 12$ $\displaystyle \small xy\leq 12$ but not $\small xy< 12$ $\displaystyle \small xy< 12$. From the second statement, we can conclude that the greatest product will result from $\small 3+3=6$ $\displaystyle \small 3+3=6$ or 9, which is less than 12.

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Example Question #2 : Dsq: Solving Inequalities

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

Statement 1: A = 8 $\displaystyle A = 8$

Statement 2: B = -8 $\displaystyle B = -8$

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.

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:

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

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

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

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

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Example Question #3 : Dsq: Solving Inequalities

True or false: $\displaystyle N$ is a positive number.

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

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

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.

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.

Explanation:

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

Therefore, Statement 2 proves $\displaystyle N$ positive, but Statement 1 only proves $\displaystyle N$ nonnegative.

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Example Question #4 : Dsq: Solving Inequalities

True or false: x < 5 $\displaystyle x < 5$

Statement 1: |x| > 5 $\displaystyle |x| > 5$

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

Possible Answers:

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

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

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

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 #5 : Dsq: Solving Inequalities

$\displaystyle x$ is a whole number.

True or false: $\displaystyle x$ is odd.

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

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

Possible Answers:

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.

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.

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

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Example Question #6 : Dsq: Solving Inequalities

True or false: x < 9 $\displaystyle x < 9$

Statement 1: |x| < 9 $\displaystyle |x| < 9$

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

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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

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

Assume Statement 2 alone. It can be rewritten as

$x^{2}< 81$ $\displaystyle x^{2}< 81$

the solution set of which is $\displaystyle -9 < x < 9$

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

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Example Question #7 : Dsq: Solving Inequalities

True or false: x > 0 $\displaystyle x > 0$

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

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

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.

Explanation:

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

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

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

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

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

$2^{x}> 1$ $\displaystyle 2^{x}> 1$

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

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

From either statement alone it follows that x>0 $\displaystyle x>0$.

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Example Question #8 : Dsq: Solving Inequalities

True or false: x < 11 $\displaystyle x < 11$

Statement 1: $x^{2} > 121$ $\displaystyle x^{2} > 121$

Statement 2: $x^{3}> 1,331$ $\displaystyle x^{3}> 1,331$

Possible Answers:

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.

BOTH statements TOGETHER are sufficient to answer the question, but NEITHER 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:

Assume Statement 1 only. Both $\displaystyle -12$ and 12 make the statement true, since $12^{2}= (-12)^{2}= 144 > 121$ $\displaystyle 12^{2}= (-12)^{2}= 144 > 121$. 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:

$x^{3}> 1,331$ $\displaystyle x^{3}> 1,331$

$\sqrt[3]{x^{3}}>\sqrt[3]{ 1,331}$ $\displaystyle \sqrt[3]{x^{3}}>\sqrt[3]{ 1,331}$

x > 11 $\displaystyle x > 11$.

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Example Question #9 : Dsq: Solving Inequalities

True or false: x > 0 $\displaystyle x > 0$

Statement 1: $x^{5}< 0$ $\displaystyle x^{5}< 0$

Statement 2: $3 ^{x}> 6^{x}$ $\displaystyle 3 ^{x}> 6^{x}$

Possible Answers:

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.

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.

Correct answer:

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, $\displaystyle x$ and $x^{5}$ $\displaystyle x^{5}$ have the same sign, and $x^{5}< 0$ $\displaystyle x^{5}< 0$ implies that x<0 $\displaystyle x< 0$.

Assume Statement 2 alone. Since $3^{x}$ $\displaystyle 3^{x}$ and $6^{x}$ $\displaystyle 6^{x}$ are both positive, we can divide both sides by $3^{x}$ $\displaystyle 3^{x}$ to yield the statement

$3^{x}>6^{x}$ $\displaystyle 3^{x}>6^{x}$

$6^{x} < 3^{x}$ $\displaystyle 6^{x} < 3^{x}$

$\frac{6^{x}}{3^{x}} <\frac{ 3^{x}}{3^{x}}$ $\displaystyle \frac{6^{x}}{3^{x}} < \frac{ 3^{x}}{3^{x}}$

$\left (\frac{6}{3} \right )^{x} <1$ $\displaystyle \left (\frac{6}{3} \right )^{x} < 1$

$2^{x}< 1$ $\displaystyle 2^{x}< 1$

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

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Example Question #10 : Dsq: Solving Inequalities

True or false: x >- 2 $\displaystyle x >- 2$

Statement 1: $x^{8} >256$ $\displaystyle x^{8} >256$

Statement 2: $x^{10} >1,024$ $\displaystyle x^{10} >1,024$

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.

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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Both statements together provide insufficient information. For example,

If x = 3 $\displaystyle x = 3$, then:

$x^{8}= 3^{8}= 6,561 > 256$ $\displaystyle x^{8}= 3^{8}= 6,561 > 256$

$x^{10}= 3^{10}= 59,049 > 1,024$ $\displaystyle x^{10}= 3^{10}= 59,049 > 1,024$

If x = -3 $\displaystyle x = -3$, then

$x^{8}= (-3)^{8}= 6,561 > 256$ $\displaystyle x^{8}= (-3)^{8}= 6,561 > 256$

$x^{10}= \left (-3 \right )^{10}= 59,049 > 1,024$ $\displaystyle x^{10}= \left (-3 \right )^{10}= 59,049 > 1,024$

Both values fit the conditions of both statements, but only one is greater than $\displaystyle -2$. The question is not answered.

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GMAT Math : Inequalities

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1 : Dsq: Solving Inequalities

Example Question #2 : Dsq: Solving Inequalities

Example Question #3 : Dsq: Solving Inequalities

Example Question #4 : Dsq: Solving Inequalities

Example Question #5 : Dsq: Solving Inequalities

Example Question #6 : Dsq: Solving Inequalities

Example Question #7 : Dsq: Solving Inequalities

Example Question #8 : Dsq: Solving Inequalities

Example Question #9 : Dsq: Solving Inequalities

Example Question #10 : Dsq: Solving Inequalities

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