GMAT Math : Equations

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Example Question #11 : Dsq: Solving Equations

Solve for :

$\left | 4x-3\right |-17=y^{2}-z^{2}$

(1) y-z=4

(2) y+z=12

Possible Answers:

Both statements TOGETHER are not sufficient

Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

Statement (1) ALONE is sufficient, but Statement (2) ALONE is not sufficient

Statement (2) ALONE is sufficient, but Statement (1) ALONE is not sufficient

Each statement ALONE is sufficient

Correct answer:

Both statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient

Explanation:

Solution

To solve for x, we need the value of $y^{2}-z^{2}$

$y^{2}-z^{2}=(y-z)(y+z)$

Therefore, we need both statements in order to solve for x.

$\left | 4x-3\right |-17= 4\times12$

$\left | 4x-3\right |-17=48$

$\left | 4x-3\right |=65$

4x-3=65 Or

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Example Question #12 : Dsq: Solving Equations

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

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

Statement 1: is a negative number.

Statement 2: $\left | x\right |>1$

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:

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

Explanation:

Assume Statement 1 alone. If is a negative number, then of the three given powers of in the set, only $x^{2}$ (the only even power) is positive. This makes $x^{2}$ the greatest of the three, thereby giving an answer to the question.

We show that Statement 2 alone is inconclusive by examining two values of whose absolute values are greater than 1 - namely, $\left \{-2, 2 \right \}$ .

Case 1: x = 2 . Then $x^{2}= 4$ and $x^{3} = 8$ , making $x^{3}$ the greatest number of the three.

Case 2: x = -2 . Then $x^{2}= 4$ and $x^{3} = -8$ , making $x^{2}$ the greatest number of the three.

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Example Question #13 : Dsq: Solving Equations

The volume of a fixed mass of gas varies inversely with the atmospheric pressure, in millibars, acting upon it, given that all other conditions remain constant.

At 12:00, a balloon was filled with exactly 100 cubic yards of helium. What its current volume?

Statement 1: The atmospheric pressure at 12:00 was 109 millibars.

Statement 2: The atmospheric pressure is now 105 millibars.

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:

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

Explanation:

You can use the following variation equation to deduce the current volume:

$V_{1} P _{1} = V_{2} P _{2}$

or, equivalently,

$V_{2} = \frac{V_{1} P _{1} }{P _{2}}$

To find the current volume $V_{2}$ , you therefore need three things - the initial volume $V_{1}$ , which is given in the body of the question; the initial pressure $P _{1}$ , which you know if you are given Statement 1; and the current pressure, $P _{2}$ , which you know if you are given Statement 2. Just substitute, and solve.

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Example Question #14 : Dsq: Solving Equations

is a rational number. True or false: N = 7

Statement 1: $N^{2} + 4N + 100 = 18N + 51$

Statement 2: |N - 14| = N

Possible Answers:

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

EITHER STATEMENT ALONE provides sufficient information to answer the question.

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

Correct answer:

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Explanation:

Assume Statement 1 alone.

$N^{2} + 4N + 100 = 18N + 51$

$N^{2} + 4N + 100- 18 N - 51 = 18N + 51 - 18 N - 51$

$N^{2} -14N +49 =0$

$(N-7)^{2} = 0$

N-7 = 0

N= 7

Assume Statement 2 alone. If |N - 14| = N , then either N - 14 = N or N - 14 =- N , so we can examine both scenarios.

Case 1:

N - 14 = N

N - 14- N = N - N

-14 = 0

This is identically false, so we dismiss this case.

Case 2:

N - 14 =- N

N - 14 + N + 14 =- N + N + 14

2N = 14

N = 7

Since each equation has only 7 as a solution, either statement alone is sufficient to identify N = 7 as a true statement.

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Example Question #15 : Dsq: Solving Equations

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

Statement 1: $3^{N} > \frac{1}{3}$

Statement 2: The arithmetic mean of 100 and is positive.

Possible Answers:

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

EITHER STATEMENT ALONE provides sufficient information to answer the question.

BOTH STATEMENTS TOGETHER do 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.

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

Correct answer:

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

Explanation:

Assume both statements are true. We show that we cannot determine for certain whether or not is positive.

Case 1: N = 0

$3^{N} = 3^{0} = 1 > \frac{1}{3}$ , satisfying the condition of Statement 1.

The arithmetic mean of 0 and 100 is half their sum, which is $\frac{0+100}{2} = 50$ , a positive number; the condition of Statement 2 is satisfied.

Case 2: N = 2

$3^{N} = 3^{2} = 9 > \frac{1}{3}$ , satisfying the condition of Statement 1.

The arithmetic mean of 2 and 100 is half their sum, which is $\frac{2+100}{2} = 51$ , a positive number; the condition of Statement 2 is satisfied.

Therefore, may or may not be positive.

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Example Question #16 : Dsq: Solving Equations

True or false: |X| = 9

Statement 1: $X^{2}+ 27 = 12X$

Statement 2: $X^{2}+6X = 27$

Possible Answers:

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

EITHER 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.

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

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

Correct answer:

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

Explanation:

The absolute value of a positive number is that number; the absolute value of a negative number is the positive opposite.

Assume Statement 1 alone.

$X^{2}+ 27 = 12X$

$X^{2}+ 27- 12X= 12X - 12X$

$X^{2}- 12X + 27= 0$

(X-9)(X-3) = 0

Either X - 9 = 0 , in which case X = 9 , and

or X-3= 0 , in which case X = 3 .

Either |X| = 9 or |X| = 3 .

Assume Statement 2 alone.

$X^{2}+6X = 27$

$X^{2}+6X - 27 = 27 - 27$

$X^{2}+6X - 27 = 0$

(X+9)(X-3) = 0

Either X + 9 = 0 , in which case X = -9 , and

or X-3= 0 , in which case X = 3 .

Either |X| = 9 or |X| = 3 .

Therefore, neither statement alone proves or disproves that |X| = 9 . But if both statements are true, then it must hold that X = 3 , since this is the only solution of both statements. Therefore, |X| = 9 can be proved false.

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Example Question #17 : Dsq: Solving Equations

is a real number. True or false: N = -3

Statement 1: |N+ 4| = 1

Statement 2: $2^{N} = 0.125$

Possible Answers:

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.

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

Correct answer:

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

Explanation:

Assume Statement 1 alone. The absolute value of a positive number is that number; the absolute value of a negative number is the positive opposite. As can be seen from substitution, there are two solutions to the equation |N+ 4| = 1 - N = -3 and N = -5

If N = -3 , the equation becomes

|N+ 4| = 1

|-3+ 4| = 1

|1| = 1 , a true statement.

If N = -5 , the equation becomes

|N+ 4| = 1

|-5+ 4| = 1

|-1| = 1 , a true statement.

Therefore, that N = -3 cannot be proved or disproved.

Assume Statement 2 alone.

$2^{N} = 0.125$ can be rewritten with the decimal expression in fraction form as

$2^{N} = \frac{1}{8}$

$2^{N} = \frac{1}{2^{3}}$

$2^{N} = 2^{-3}$

Since if two powers of the same number are equal, the exponents are equal, it follows that N = -3 .

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Example Question #18 : Dsq: Solving Equations

True or false: | N | = 5

Statement 1: N (N+10 ) = 5 (2N+5)

Statement 2: | N + 10 | = 15

Possible Answers:

BOTH STATEMENTS TOGETHER do NOT provide 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 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.

EITHER STATEMENT ALONE provides sufficient information to answer the question.

Correct answer:

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. The solutions of the equation can be found as follows:

N (N+10 ) = 5 (2N+5)

$N ^{2}+10 N = 10N+25$

$N ^{2}+10 N - 10 N = 10N+25 - 10 N$

$N ^{2}=25$

$N = \pm \sqrt{25} = \pm 5$

The absolute value of a positive number is that number; the absolute value of a negative number is the positive opposite. If N = 5 , then | N | = 5 . If N = - 5 , then also. Therefore, while the value of cannot be determined for certain,either way, is a true statement.

Assume Statement 2 alone. By substitution, we can see two values of that make this true:

N = 5

| N + 10 | = 15

| 5 + 10 | = 15

|15| = 15 , which is true.

N = -25

| N + 10 | = 15

| -25 + 10 | = 15

|-15| = 15 , which is also true.

One of these two solutions has absolute value 5, but the other does not. This makes it indefinite whether | N | = 5 .

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GMAT Math : Equations

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #11 : Dsq: Solving Equations

Example Question #12 : Dsq: Solving Equations

Example Question #13 : Dsq: Solving Equations

Example Question #14 : Dsq: Solving Equations

Example Question #15 : Dsq: Solving Equations

Example Question #16 : Dsq: Solving Equations

Example Question #17 : Dsq: Solving Equations

Example Question #18 : Dsq: Solving Equations

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