GMAT Math : Algebra

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Example Question #10 :Dsq: Understanding Absolute Value

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

Statement 1: 2 N + 7 < 17

Statement 2: 4N - 17 > -37

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.

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.

Correct answer:

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

Explanation:

If $\left | N\right | < 5$ , then, by definition, -5 < N < 5 - that is, both N < 5 and N > -5 .

If Statement 1 is true, then

2 N + 7 < 17

2 N < 10

N < 5

Statement 1 alone does not answer the question, as N < 5 follows, but not necessarily N > -5 .

If Statement 2 is true, then

4N - 17 > -37

4N >-20

N > -5

Statement 2 alone does not answer the question, as N > -5 follows, but not necessarily N < 5 .

Ifbothstatements are true, then N > -5 and N < 5 both follow, and -5 < N < 5 , meaning that $\left | N \right | < 5$ .

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Example Question #11 :Dsq: Understanding Absolute Value

Of distinct integers a, b, c , which is the greatest of the three?

Statement 1: |b| <a < |c|

Statement 2: |a | + |b| = |c|

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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

The two statements together are insufficient.

For example, let a = 8, b=6 . Then, from Statement 2,

|c| = |a| + |b| = |6 | + |8| = 6 + 8 = 14

Therefore, either c = - 14 or c = 14 .

In either case, Statement 2 is shown to be true, since

6 < 8 < 14

and

|b| <a < |c|

But if c = - 14 , thenis the greatest of the three. If c = 14 , thenis the greatest. Therefore,the two statements together are not enough.

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Example Question #12 :Dsq: Understanding Absolute Value

Of distinct integers a, b, c , which is the greatest of the three?

Statement 1: |a| < b < |c|

Statement 2:andare negative.

Possible Answers:

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.

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.

Correct answer:

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

Explanation:

Statement 1 alone gives insufficient information.

Case 1: a=4 , b= 5, c =6

|a|< b < |c|

|4|< 5 < |6|

4< 5 < 6 , which is true.

Case 2: a=-4 , b= 5, c =-6

|a|< b < |c|

|-4|< 5 < |-6|

4< 5 < 6 , which is true.

But in the first case,is the greatest of the three. In the second,is the greatest.

Statement 2 gives insuffcient information, since no information is given about the sign of.

Assume both statements to be true. $|a| \ge 0$ , and from Statement 1, b > |a| ; by transitivity, b > 0 . From Statement 2, a < 0, c < 0 . This makesthe greatest of the three.

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Example Question #13 :Dsq: Understanding Absolute Value

Let A,B,C be any three (not necessarily distinct) integers.

At how many points does the graph of the function

f (x)= A |x-B | + C

intersect the- axis?

Statement 1:andare nonzero integers of opposite sign.

Statement 2:andare nonzero integers of opposite sign.

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:

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

Explanation:

To determine the point(s), if any, at which the graph of a functionintersects the- axis, set f(x)= 0 and solve for.

f (x)= A |x-B | + C

A |x-B | + C = 0

A |x-B | = - C

$|x-B | = - \frac{C}{A}$

At this point, we can examine the equation. Since the absolute value of a number must be nonnegative, the sign of $- \frac{C}{A}$ tells us how many solutions exist to this equation. If $- \frac{C}{A} < 0$ , there is no solution, and therefore, the graph ofdoes not intersect the- axis. If $- \frac{C}{A} = 0$ , then there is one solution, and, therefore, the graph ofintersects the- axis at exactly one point. If $- \frac{C}{A} > 0$ , then there are two solutions, and, therefore, the graph ofintersects the- axis at exactly two points.

To determine the sign of $- \frac{C}{A}$ , we need to whether the signs of bothandare like or unlike, or that C = 0 . Either statement alone eliminates the possibility that C = 0 , but neither alone gives the signs of bothand. However, if both statements are assumed, then, sinceandhave the opposite sign as, they have the same sign. This makes $\frac{C}{A} > 0$ and $- \frac{C}{A} < 0$ , so the graph ofcan be determined to not cross the- axis at all.

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Example Question #14 :Dsq: Understanding Absolute Value

Which, if either, is the greater number:or?

Statement 1: M = |N - 10|

Statement 2: N = |M + 10|

Possible Answers:

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.

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:

Statement 1 alone gives insufficient information, as is seen in these two cases:. For example, if N = 15 , then

M = |N - 10|

M = |15 - 10| = |5| = 5

M < N

However, if N = 0 , then

M = |N - 10|

M = |0 - 10| = |-10| = 10

M > N

Therefore, it is not clear which, if either, ofandis greater.

Now assume Statement 2 alone.

Ifis negative, then, which, being an absolute value of a number, must be nonnegative, is the greater number. Ifis positive, then so is M + 10 , so

N = |M + 10| = M + 10 .

Therefore,

N > M .

is the greater number in either case.

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Example Question #15 :Dsq: Understanding Absolute Value

Let A,B,C be any three (not necessarily distinct) integers.

At how many points does the graph of the function

f (x)= A |x-B | + C

intersect the- axis?

Statement 1: B < 0

Statement 2: C< 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:

BOTH statements TOGETHER are insufficient to answer the question.

Explanation:

To determine the point(s), if any, at which the graph of a functionintersects the- axis, set f(x)= 0 and solve for.

f (x)= A |x-B | + C

A |x-B | + C = 0

A |x-B | = - C

$|x-B | = - \frac{C}{A}$

At this point, we can examine the equation. For a solution to exist, since the absolute value of a number must be nonnegative, it must hold that $- \frac{C}{A} \geq 0$ . This happens ifandare of opposite sign, or if C = 0 . However, Statement 2 tells us that C< 0 , and neither statement tells us the sign of. The two statements together provide insufficient information.

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GMAT Math : Algebra

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #10 :Dsq: Understanding Absolute Value

Example Question #11 :Dsq: Understanding Absolute Value

Example Question #12 :Dsq: Understanding Absolute Value

Example Question #13 :Dsq: Understanding Absolute Value

Example Question #14 :Dsq: Understanding Absolute Value

Example Question #15 :Dsq: Understanding Absolute Value

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