GMAT Math : Algebra

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Example Question #9 :Dsq: Understanding Factoring

Simplify: $\frac{-3x + 3}{x^2 - 1}$

Possible Answers:

$\frac{-3}{x-1}$

$\frac{-3x}{x-1}$

$\frac{3}{x-1}$

$\frac{-3}{x+1}$

$\frac{-3x}{x+1}$

Correct answer:

$\frac{-3}{x+1}$

Explanation:

In order to simplify, we must first pull out the largest common factor of each term in the numerator, -3:

$\frac{-3x+3}{x^2 - 1} = \frac{-3(x-1)}{x^2-1}$

We then recognize that the denominator is a difference of squares:

$\frac{-3(x-1)}{x^2-1} = \frac{-3(x-1)}{(x-1)(x+1)}$

We can therefore cancel the (x-1) terms and are left with:

$\frac{-3}{x+1}$

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Example Question #1 :Dsq: Understanding Absolute Value

Given that x + y = 10 , evaluate x - y 。

1) $x^{2} - y^{2} = 40$

2) xy = 21

Possible Answers:

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

BOTH statements TOGETHER are NOT sufficient to answer the question.

EITHER Statement 1 or Statement 2 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.

Correct answer:

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

Explanation:

$x^{2} - y^{2} = (x + y) (x - y)$ ,

so, if we know x + y = 10 and $x^{2} - y^{2} = 40$ , then the above becomes

40 = 10 (x - y)

and $x - y = 40 \div 10 = 4$

If we know x + y = 10 and xy = 21 , then we need two numbers whose sum is 10 and whose product is 21; by inspection, these are 3 and 7. However, we do not know whether x = 3 and y = 7 or vice versa just by knowing their sum and product. Therefore, either x - y = 7 - 3 = 4 , or x - y = 3 - 7 = -4 。

The answer is that Statement 1 alone is sufficient, but not Statement 2.

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Example Question #2 :Dsq: Understanding Absolute Value

Using the following statements, Solve for。

x=|A-B| (read asequals the absolute value ofminus)

1. B-A= 4

2. A+B=10

Possible Answers:

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.

Statements (1) and (2) TOGETHER are NOT sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Correct answer:

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

Explanation:

This question tests your understanding of absolute value. You should know that

|B-A|=|A-B| since we are finding the absolute value of the difference. We can prove this easily. Since B-A= -(A-B) , we know their absolute values have to be the same.

Therefore, statement 1 alone is enough to solve for。我们得到 x=4 。

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Example Question #3 :Dsq: Understanding Absolute Value

Is $\left | a-b \right |<\left | a-c \right |?$

(1) a>0

(2) $\left | b \right |<\left | c \right |$

Possible Answers:

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.

Statements (1) and (2) TOGETHER are NOT sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

Correct answer:

Statements (1) and (2) TOGETHER are NOT sufficient.

Explanation:

For statement (1), since we don’t know the value ofand, we have no idea about the value of $\left | a-b \right |$ and $\left | a-c \right |$ 。

For statement (2), since we don’t know the sign ofand, we cannot compare $\left | a-b \right |$ and $\left | a-c \right |$ 。

Putting the two statements together, if a-b>0 and a-c>0 , then $\left | a-b \right |<\left | a-c \right |$ 。

But if a-b<0 and a-c<0 , then $\left | a-b \right |>\left | a-c \right |$ 。

Therefore, we cannot get the only correct answer for the questions, suggesting that the two statements together are not sufficient. For this problem, we can also plug in actual numbers to check the answer.

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Example Question #4 :Dsq: Understanding Absolute Value

Is nonzero numberpositive or negative?

Statement 1: N + |N| = 0

Statement 2: $N^{2} + 5N < 0$

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.

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If we assume that N + |N| = 0 , then it follows that:

N + |N| - |N| = 0- |N|

N = - |N|

Since we know $N \neq 0$ , we know |N| is positive, and - |N| andare negative.

If we assume that $N^{2} + 5N < 0$ , then it follows that:

$N^{2} + 5N -N^{2} < 0-N^{2}$

$5N < -N^{2}$

$N < -\frac{N^{2}}{5}$

Since we know $N \neq 0$ , we know $N^{2}$ is positive. $\frac{N^{2}}{5}$ is also positive and $-\frac{N^{2}}{5}$ is negative; sinceis less than a negative number,is also negative.

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Example Question #5 :Dsq: Understanding Absolute Value

True or false: N < 7

Statement 1: $\left | N\right | <7$

Statement 2: $N^{2} < 49$

Possible Answers:

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.

Correct answer:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

Statement 1 and Statement 2 are actually equivalent.

If $\left | N\right | <7$ , then either -7 < N < 7 by definition.

If $N^{2} < 49$ , then either -7 < N < 7 。

From either statement alone, it can be deduced that N < 7 。

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Example Question #6 :Dsq: Understanding Absolute Value

is a real number. True or false: N > 10

Statement 1: $\left | N\right | > 10$

Statement 2: $N^{2} > 100$

Possible Answers:

BOTH statements TOGETHER are insufficient 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 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.

Correct answer:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

Statement 1 and Statement 2 are actually equivalent.

If $\left | N\right | > 10$ , then either N > 10 or N < -10 by definition.

If $N^{2} > 100$ , then either N > 10 or N < -10 。

The correct answer is that the two statements together are not enough to answer the question.

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Example Question #7 :Dsq: Understanding Absolute Value

is a real number. True or false: $\left | N\right | < 5$

Statement 1: $\left | N - 1\right | < 4$

Statement 2: $\left | N +1 \right | <6$

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.

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.

Correct answer:

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

Explanation:

If $\left | N\right | < 5$ , then, by definition, -5 < N < 5 。

If Statement 1 is true, then

$\left | N - 1\right | < 4$

-4 < N - 1 < 4

-3 < N < 5 ,

somust be in the desired range.

If Statement 2 is true, then

$\left | N +1 \right | <6$

-6 < N +1 <6

-7< N <5

andis not necessarily in the desired range.

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Example Question #8 :Dsq: Understanding Absolute Value

is a real number. True or false: N > 5

Statement 1: $\left | N\right | > 5$

Statement 2: $N^{3} > 125$

Possible Answers:

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.

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:

If $\left | N\right | > 5$ , then we can deduce only thateither N > 5 or N < -5 。Statement 1 alone does not answer the question.

If $N^{3} > 125$ , thenmust be positive, as no negative number can have a positive cube. The positive numbers whose cubes are greater than 125 are those greater than 5. Therefore, Statement 2 alone proves that N > 5 。

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Example Question #9 :Dsq: Understanding Absolute Value

is a real number. True or false: $\left | N\right | < 6$

Statement 1: $N^{2} > 36$

Statement 2: $N^{3} > 216$

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:

EITHER statement ALONE is sufficient to answer the question.

Explanation:

If $\left | N\right | < 6$ , then, by definition, -6 < N < 6 。

If Statement 1 holds, that is, if $N^{2} > 36$ , one of two things happens:

Ifis positive, then $N > \sqrt{36} = 6$ 。

Ifis negative, then $N < - \sqrt{36} = -6$ 。

$\left | N\right | < 6$ is a false statement.

If Statement 2 holds, that is, if $N^{3} > 216$ , we know thatis positive, and

$N >\sqrt[3]{ 216} = 6$

$\left | N\right | < 6$ is a false statement.

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #9 :Dsq: Understanding Factoring

Example Question #1 :Dsq: Understanding Absolute Value

Example Question #2 :Dsq: Understanding Absolute Value

Example Question #3 :Dsq: Understanding Absolute Value

Example Question #4 :Dsq: Understanding Absolute Value

Example Question #5 :Dsq: Understanding Absolute Value

Example Question #6 :Dsq: Understanding Absolute Value

Example Question #7 :Dsq: Understanding Absolute Value

Example Question #8 :Dsq: Understanding Absolute Value

Example Question #9 :Dsq: Understanding Absolute Value

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