If x, y and z are integers and x – y – z

Question

If x, y and z are integers and x – y – z < 0, is z > 1?

(1) x - y > 1 - z
(2) y - x < -2

*kudos for all correct solutions

ASIDE: Given the issues with my first posting of the question, I should upgrade this question to 800!!

broall · Answer

We have $x 1 - z \implies x > y +1 -z \implies y+z > y+ 1 -z \implies 2z > 1 \implies z > \frac{1}{2}$. Hence we have $z=1$, $z<1$ or $z>1$. Insufficient. (2) $y-x<-2 \implies y+2 < x < y+ z \implies z > 2 > 1$. Sufficient. The answer is B.

Mo2men · Answer

If x, y and z are integers and x – y – z < 0, is z > 1?

Alternative method by plugging some values. It is somehow cumbersome for statement 1.

Starting with Statement 2:

2) y - x < -2

Let y-x=-2.1 < -2

Apply in equation in the stem:

-2.1 - z <0............z>2.1>1...........Answer is Yes . You spot that if we increase the magnitude of (y-x), z is getting bigger. If you do not notice, check other numbers.

Let y-x = -3 < -2

Apply in equation in the stem:

-3 - z <0............z>3>1...........Answer is Yes. You can check other points and you will find the same.

Sufficient

(1) x - y > 1 - z

Let x-y=3.... We need to maintain the inequality in the stem true also

3>1 - z

z might be 4.... check in inequality in stem 3-4 <0...........So z > 1

z might be INTEGER GREATER than 1. So it could be 2, 3,4,..etc

Answer is always YES.........Please read below posts to check proof.

Sufficient

Answer: D

Note: It is really cumbersome to plug in numbers especially for number 1

Mo2men · Answer

If x, y and z are integers and x – y – z < 0, is z > 1?

Alternative method by plugging some values. It is somehow cumbersome for statement 1.

Starting with Statement 2:

2) y - x < -2

Let y-x=-2.1 < -2

Apply in equation in the stem:

-2.1 - z <0............z>2.1>1...........Answer is Yes . You spot that if we increase the magnitude of (y-x), z is getting bigger. If you do not notice, check other numbers.

Let y-x = -3 < -2

Apply in equation in the stem:

-3 - z <0............z>3>1...........Answer is Yes. You can check other points and you will find the same.

Sufficient

(1) x - y > 1 - z

Let x-y=3.... We need to maintain the inequality in the stem true also

3>1 - z

z might be 4.... check in inequality in stem 3-4 <0...........So z > 1

z might be INTEGER GREATER than 1. So it could be 2, 3,4,..etc

Answer is always YES.........Please read below posts to check proof.

Sufficient

Answer: D

Note: It is really cumbersome to plug in numbers especially for number 1

BrentGMATPrepNow · Answer

Good solution Mo2men. Only one small glitch z cannot be 1/2, since x, y and z are integers.

Cheers,
Brent

Mo2men · Answer

Dear Brent, Good catch from you. But I really could not come up with numbers to prove insufficient. All what we know that z> 1/2. If z =1 & x-y=3 3 – 1 < 0........Not valid 3 + 1> 1 ........Valid If z =1 & x-y=- 3 -3 -1 <0....valid - 3 + 1 > 1...Not valid Z can't be viable number to test as it violates either conditions in the stem of fact 1 If z =4 & x-y=3 3 – 4 < 0........valid 3 + 4> 1 ........Valid If z =5 & x-y=-3 - 3 – 5 < 0........valid - 3 +5> 1 ........Valid so Z must be greater than 1 So It seems A is sufficient too. What do you think?

BrentGMATPrepNow · Answer

I think you're right. Given : x – y – z < 0 Target question : Is z > 1 (1) x - y > 1 - z From statement 1 (and the given inequality), we learn that z > 1/2. So, we might (incorrectly) conclude that it could be the case that z = 1 or z = 2, in which case, we get different answers to the target question. HOWEVER, if we try to come up with values for x, y and z that demonstrate this, we find that we have a problem. If z = 1 , then we can plug this value into our two inequalities. For the statement 1 inequality, we get x - y > 1 - 1 Simplify to get: x - y > 0 For the given inequality, we get x – y – 1 < 0 Simplify to get: x - y < 1 When we combine the two inequalities , we get: 0 < x - y < 1 In other words, the difference between x and y is a fractional value BETWEEN 0 and 1. This is IMPOSSIBLE, since it's given that x and y are integers. So, it cannot be the case that z = 1 Since we already know that z > 1/2, we can conclude that it's possible that z = 2, z = 3, z = 4, etc. For example, consider these situations: Case a: x = 0, y = 0 and z = 2. In this case, z IS greater than 1 Case b: x = 0, y = 0 and z = 3. In this case, z IS greater than 1 Case c: x = 0, y = 0 and z = 4. In this case, z IS greater than 1 Case d: x = 0, y = 0 and z = 5. In this case, z IS greater than 1 etc.. So, it turns out that the correct answer is actually D (both statements are sufficient) Sorry for not knowing the correct answer when I first posted the question. It's even harder than I first imagined!! Cheers, Brent

Mo2men · Answer

Dear Brent,

You do not need to be sorry. You always provide us with genuine question and answers.

Thanks for your keen support and help

I have never asked before to get kudos but I need a kudos from you :wink:

BrentGMATPrepNow · Answer

You deserve a bucket of kudos for catching that!! :-D

Cheers, Brent

arh5451 · Answer

Can you please post the answer choices i don't understand without them?

Bunuel · Answer

Hi,

This is a data sufficiency question. Options for DS questions are always the same.

The data sufficiency problem consists of a question and two statements, labeled (1) and (2), in which certain data are given. You have to decide whether the data given in the statements are sufficient for answering the question. Using the data given in the statements, plus your knowledge of mathematics and everyday facts (such as the number of days in July or the meaning of the word counterclockwise), you must indicate whether—

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

I suggest you to go through the following post  ALL YOU NEED FOR QUANT .

Hope this helps.

arh5451 · Answer

Thank you for the response. In that case is this how you can solve? Or am I crazy?

I believe you can use the approach z=2 as the test case. Since we want to know if it can be larger than 1 and it must be an integer. Essentially we can rephrase this is it possible that z = 2?

Case 1 test:
x-y-z > 0
or
x-y-2 = 0 --> x - y > 2

x - y > 1 - z
x - y > 1 - 2 --> x - y > -1

if x and y are 5 and 3 then its possible. True

Case 2 test:

x-y >= 2

y-x < -2

if x = 6 and y = 3 then 6-3 > 2 and 3-6 < -2 so its True.

Case 1 and Case 2:

x - y > 2
x - y > -1
y - x < -2

if x = 6 and y = 3

D)

gmatophobia · Answer

We can use the concept of a number line to solve this question. Attached video solution to the question -

https://www.youtube.com/watch?v=-9Lmn35QLVA

If x, y and z are integers and x – y – z < 0, is

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions