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# If x > 2 and y > -1, then which of the following statements must be tr

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Math Expert
Joined: 02 Sep 2009
Posts: 57074
If x > 2 and y > -1, then which of the following statements must be tr  [#permalink]

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25 Jun 2019, 02:06
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Difficulty:

55% (hard)

Question Stats:

57% (01:51) correct 43% (01:30) wrong based on 88 sessions

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If $$x>2$$ and $$y>-1$$, then which of the following statements must be true?

A. $$xy>-2$$
B. $$-x<2y$$
C. $$xy<-2$$
D. $$-x>2y$$
E. $$x<-2y$$

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Re: If x > 2 and y > -1, then which of the following statements must be tr  [#permalink]

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25 Jun 2019, 06:52
Bunuel wrote:
If $$x>2$$ and $$y>-1$$, then which of the following statements must be true?

A. $$xy>-2$$
B. $$-x<2y$$
C. $$xy<-2$$
D. $$-x>2y$$
E. $$x<-2y$$

Strategy: Try to find exceptions for each option!!

A. $$xy>-2$$
--> x = 10, y = -0.5 xy = -5 (<-2)
-- NO

B. $$-x<2y$$
--> Possible values of 2y are -1.9, -1, 0, 2, 5, 100;
Possible values of -x are -2, -2.5, -3, -4 etc
-- YES

C. $$xy<-2$$
--> x = 3, y = -0.5 xy = -1.5 (>-2)
-- NO

D. $$-x>2y$$
--> Possible values of 2y are -1.9, -1, 0, 2, 5, 100;
Possible values of -x are -2, -2.5, -3, -4 etc
-- NO

E. $$x<-2y$$
--> x = 100, y = 0
--> (x, -2y) = (100, 0)
-- NO

IMO Option B

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If x > 2 and y > -1, then which of the following statements must be tr  [#permalink]

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Updated on: 25 Jun 2019, 10:36
1
1
Nice question!
Constraint x > 2 so we know x is always +ve
Also, y > -1 so we know y is -ve (non-integer),+ve or 0

(a) xy > -2 —> +ve•+ve >-2 (true), +ve•-ve(non-integer) >-2 (not always true) so this option is not necessarily true

(b) -x < 2y —> x > -2y ,now the LHS of the inequality is always +ve and RHS y can be zero(0),-ve(non-integer) or +ve( but ends up being negative eg. -2•(2)=-4)
(Always true in all cases )

(c) xy < -2 —> +ve•+ve >0 so when xy is +ve (not true)

(d) -x > 2y —> Here x is -ve and can never be greater than 2y when y=+ve (not true)

(e) x <-2y —> not true when y is +ve

Posted from my mobile device

Originally posted by Staphyk on 25 Jun 2019, 09:06.
Last edited by Staphyk on 25 Jun 2019, 10:36, edited 1 time in total.
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Re: If x > 2 and y > -1, then which of the following statements must be tr  [#permalink]

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25 Jun 2019, 10:15
The product xy can be anything, because x can be 1 trillion, say, and y can be negative (so xy can be very very negative) or positive (so xy can be very very positive). So A and C do not need to be true. If you then notice that D and E say the same thing (multiply D by -1 on both sides, reversing the inequality when you do, and you get E) then you can be sure neither of them can be the right answer, because if one were always true, the other would be too, and then the question would have two different right answers, which is impossible. So the answer has to be B.

We can prove that though: if x > 2, then -x < -2. If y > -1, then 2y > -2. So putting these two inequalities together, -x < -2 < 2y, and -x < 2y.
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Re: If x > 2 and y > -1, then which of the following statements must be tr  [#permalink]

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01 Jul 2019, 18:02
2
Bunuel wrote:
If $$x>2$$ and $$y>-1$$, then which of the following statements must be true?

A. $$xy>-2$$
B. $$-x<2y$$
C. $$xy<-2$$
D. $$-x>2y$$
E. $$x<-2y$$

We can let x = 3 and y = 0, we see that choices A and B are true while the other ones are false. Now let’s let x = 3 and y = -0.9, we see that xy = -2.7 which is not greater than -2. However, -x = -3, which is less than 2y = -1.8. So only B is the correct answer.

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Re: If x > 2 and y > -1, then which of the following statements must be tr   [#permalink] 01 Jul 2019, 18:02
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