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If x, y and z are integers and x – y – z < 0, is [#permalink]
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Updated on: 22 Mar 2017, 10:52
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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!!
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Brent Hanneson – Founder of gmatprepnow.com
Originally posted by GMATPrepNow on 21 Mar 2017, 07:49.
Last edited by GMATPrepNow on 22 Mar 2017, 10:52, edited 2 times in total.



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Re: If x, y and z are integers and x – y – z < 0, is [#permalink]
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21 Mar 2017, 08:48
GMATPrepNow wrote: 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 We have \(x<y+z\) (1) \(xy > 1  z \implies x > y +1 z \implies y+z > y+ 1 z \implies 2z > 1 \implies z > \frac{1}{2}\). Hence we have \(z=1\), \(z<1\) or \(z>1\). Insufficient. (2) \(yx<2 \implies y+2 < x < y+ z \implies z > 2 > 1\). Sufficient. The answer is B.
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If x, y and z are integers and x – y – z < 0, is [#permalink]
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22 Mar 2017, 03:46
If x, y and z are integers and x – y – z < 0, is z > 1?
(1) x  y > 1  z
Rearranging:
x  y +z > 1 x – y – z < 0  Subtracting 2 inequality AS LONG AS the signs are in OPPOSITE directions. 0+0 +2z > 1
z>1/2......
z CAN'T be 1 as it violates one of the two conditions. Test other INTEGER numbers GREATER THAN 1
z might be 2, 3,4,..etc
Answer is always YES.........Please read below posts to check proof.
Sufficient
(2) y  x < 2
x – y – z < 0 y  x < 2  Adding two inequalities AS LONG AS the signs in SAME directions 0 + 0 z< 2
z <2 ....... multiply by 1 z > 2>1
Sufficient
Answer: D



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If x, y and z are integers and x – y – z < 0, is [#permalink]
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22 Mar 2017, 04:53
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 yx=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 (yx), z is getting bigger. If you do not notice, check other numbers. Let yx = 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 xy=3.... We need to maintain the inequality in the stem true also
3>1  z
z might be 4.... check in inequality in stem 34 <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



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Re: If x, y and z are integers and x – y – z < 0, is [#permalink]
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22 Mar 2017, 08:04
Mo2men wrote: z might be 1/2....We need to maintain the inequality in the stem true also 3/4 <0.......Sp z <1
Good solution Mo2men. Only one small glitch z cannot be 1/2, since x, y and z are integers. Cheers, Brent
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Re: If x, y and z are integers and x – y – z < 0, is [#permalink]
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22 Mar 2017, 08:16
GMATPrepNow wrote: Mo2men wrote: z might be 1/2....We need to maintain the inequality in the stem true also 3/4 <0.......Sp z <1
Good solution Mo2men. Only one small glitch z cannot be 1/2, since x, y and z are integers. Cheers, Brent 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 & xy=3 3 – 1 < 0........Not valid 3 + 1> 1 ........Valid If z =1 & xy= 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 & xy=3 3 – 4 < 0........valid 3 + 4> 1 ........Valid If z =5 & xy=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?



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Re: If x, y and z are integers and x – y – z < 0, is [#permalink]
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22 Mar 2017, 10:45
Mo2men wrote: But I really could not come up with numbers to prove insufficient. All what we know that z> 1/2.
I think you're right. Given: x – y – z < 0 Target question: Is z > 1(1) x  y > 1  zFrom 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  1Simplify to get: x  y > 0For the given inequality, we get x – y – 1 < 0 Simplify to get: x  y < 1When we combine the two inequalities, we get: 0 < x  y < 1In 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 = 1Since 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 1Case b: x = 0, y = 0 and z = 3. In this case, z IS greater than 1Case c: x = 0, y = 0 and z = 4. In this case, z IS greater than 1Case d: x = 0, y = 0 and z = 5. In this case, z IS greater than 1etc.. 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
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Re: If x, y and z are integers and x – y – z < 0, is [#permalink]
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22 Mar 2017, 10:52
GMATPrepNow wrote: Mo2men wrote: But I really could not come up with numbers to prove insufficient. All what we know that z> 1/2.
I think you're right. Given: x – y – z < 0 Target question: Is z > 1(1) x  y > 1  zFrom 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  1Simplify to get: x  y > 0For the given inequality, we get x – y – 1 < 0 Simplify to get: x  y < 1When we combine the two inequalities, we get: 0 < x  y < 1In 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 = 1Since 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 1Case b: x = 0, y = 0 and z = 3. In this case, z IS greater than 1Case c: x = 0, y = 0 and z = 4. In this case, z IS greater than 1Case d: x = 0, y = 0 and z = 5. In this case, z IS greater than 1etc.. 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 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



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If x, y and z are integers and x – y – z < 0, is [#permalink]
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22 Mar 2017, 10:58
Mo2men wrote: 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 You deserve a bucket of kudos for catching that!! Cheers, Brent
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