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# Is xy <= 6 ? (1) 1 <= x <= 2 and 4 <= y <= 6 (2) x + y = 7

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Math Expert
Joined: 02 Sep 2009
Posts: 43377
Is xy <= 6 ? (1) 1 <= x <= 2 and 4 <= y <= 6 (2) x + y = 7 [#permalink]

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16 Dec 2017, 00:46
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Difficulty:

45% (medium)

Question Stats:

63% (01:09) correct 37% (01:34) wrong based on 52 sessions

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Is $$xy \leq 6$$ ?

(1) $$1 \leq x \leq 2$$ and $$4 \leq y \leq 6$$
(2) x + y = 7
[Reveal] Spoiler: OA

_________________
Manager
Joined: 24 Nov 2016
Posts: 151
Is xy <= 6 ? (1) 1 <= x <= 2 and 4 <= y <= 6 (2) x + y = 7 [#permalink]

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16 Dec 2017, 08:56
Bunuel wrote:
Is $$xy \leq 6$$ ?

(1) $$1 \leq x \leq 2$$ and $$4 \leq y \leq 6$$
(2) x + y = 7

Is $$xy \leq 6$$? If x and y are opposite signs, then xy is ≤ 6; If they are the same sign, then we must solve for x and y.

(1) $$1 \leq x \leq 2$$ and $$4 \leq y \leq 6$$. If we multiply both inequalities we have: $$4 \leq xy \leq 12$$, then xy could be ≥ 6 or xy ≤ 6, insufficient.

(2) $$x + y = 7$$. If x = 3 and y = 4, xy = 12, condition is false. If x = 1 and y = 6, xy = 6, condition is true. Both cases, x and y could be opposite signs or not.. insufficient.

(3) Combining both: (1) x and y are the same signs (positive) bound by the inequality, and (2) their sum has to be equal to 7.

So if x = 1 y = 6, x + y = 7 and xy = 6 ≤ 6. But if x = 2 y = 5, x + y = 7 then xy = 10 ≥ 6. We have more than one possibility, insufficient.

Last edited by exc4libur on 18 Dec 2017, 02:05, edited 2 times in total.
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Re: Is xy <= 6 ? (1) 1 <= x <= 2 and 4 <= y <= 6 (2) x + y = 7 [#permalink]

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16 Dec 2017, 11:08
1
KUDOS
exc4libur wrote:
Bunuel wrote:
Is $$xy \leq 6$$ ?

(1) $$1 \leq x \leq 2$$ and $$4 \leq y \leq 6$$
(2) x + y = 7

Is $$xy \leq 6$$? If x and y are opposite signs, then xy is ≤ 6; If they are the same sign, then we must solve for x and y.

(1) $$1 \leq x \leq 2$$ and $$4 \leq y \leq 6$$. If we multiply both inequalities we have: $$4 \leq xy \leq 12$$, then xy could be ≥ 6 or xy ≤ 6, insufficient.

(2) $$x + y = 7$$. If x = 3 and y = 4, xy = 12, condition is false. If x = 3 and y = -4, xy = -12, condition is true. Both cases, x and y could be opposite signs or not.. insufficient.

(3) Combining both: (1) x and y are the same signs (positive), and (2) their sum is equal to 7. There are only 2 solutions for x that meet the inequality: x = 1 or x = 2. And to meet the (2) statement, y has to be y = 6 or y = 5. Thus, xy is either (1)(6) = 6 or (2)(5) = 10, which both are ≥ 6, sufficient.

Hi exc4libur

highlighted portion is not correct. if x=3 & y =-4 then x+y=-1 which is not possible here.
Manager
Joined: 24 Nov 2016
Posts: 151
Re: Is xy <= 6 ? (1) 1 <= x <= 2 and 4 <= y <= 6 (2) x + y = 7 [#permalink]

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16 Dec 2017, 11:20
Quote:
Hi exc4libur

highlighted portion is not correct. if x=3 & y =-4 then x+y=-1 which is not possible here.

Hi niks, thanks!! I got distracted, but its correct now. regds.
VP
Joined: 26 Mar 2013
Posts: 1382
Re: Is xy <= 6 ? (1) 1 <= x <= 2 and 4 <= y <= 6 (2) x + y = 7 [#permalink]

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17 Dec 2017, 16:42
1
KUDOS
exc4libur wrote:
Bunuel wrote:
Is $$xy \leq 6$$ ?

(1) $$1 \leq x \leq 2$$ and $$4 \leq y \leq 6$$
(2) x + y = 7

Is $$xy \leq 6$$? If x and y are opposite signs, then xy is ≤ 6; If they are the same sign, then we must solve for x and y.

(1) $$1 \leq x \leq 2$$ and $$4 \leq y \leq 6$$. If we multiply both inequalities we have: $$4 \leq xy \leq 12$$, then xy could be ≥ 6 or xy ≤ 6, insufficient.

(2) $$x + y = 7$$. If x = 3 and y = 4, xy = 12, condition is false. If x = 1 and y = 6, xy = 6, condition is true. Both cases, x and y could be opposite signs or not.. insufficient.

(3) Combining both: (1) x and y are the same signs (positive), and (2) their sum is equal to 7. There are only 2 solutions for x that meet the inequality: x = 1 or x = 2. And to meet the (2) statement, y has to be y = 6 or y = 5. Thus, xy is either (1)(6) = 6 or (2)(5) = 10, which both are ≥ 6, sufficient.

Hi,

Based on what you have provided, the answer should be E.

x =1 & y = 6.........xy ≤ 6.............Answer is Yes

x =2 & y = 5.........xy ≤ 10.............Answer is NO

Also small comment. The highlighted is not correct. You assumed that x and y are integers but x could be 1.9 & y could be 5.1. Nothing in the questions says that those are integers.
Intern
Joined: 12 Nov 2017
Posts: 5
Re: Is xy <= 6 ? (1) 1 <= x <= 2 and 4 <= y <= 6 (2) x + y = 7 [#permalink]

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17 Dec 2017, 19:30
The question is, Is xy<= 6? Hence, (1)(6) = 6 is a YES and (2)(5) = 10 is a NO. Answer should be E.
Manager
Joined: 24 Nov 2016
Posts: 151
Re: Is xy <= 6 ? (1) 1 <= x <= 2 and 4 <= y <= 6 (2) x + y = 7 [#permalink]

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18 Dec 2017, 02:06
Mo2men wrote:
exc4libur wrote:
Bunuel wrote:
Is $$xy \leq 6$$ ?

Done, thank you Mo
Re: Is xy <= 6 ? (1) 1 <= x <= 2 and 4 <= y <= 6 (2) x + y = 7   [#permalink] 18 Dec 2017, 02:06
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