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If x>0, y<0, and a>b>0, then which of the following is (are) positive?

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If x>0, y<0, and a>b>0, then which of the following is (are) positive? [#permalink]

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New post 14 Jul 2017, 01:07
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If x>0, y<0, and a>b>0, then which of the following is (are) positive?

Ⅰ. ax+by Ⅱ. ax-by Ⅲ. by-ax

A. Ⅰonly
B. Ⅱ only
C. Ⅲ only
D.Ⅰ& Ⅱ only
E. Ⅱ & Ⅲ only

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If x>0, y<0, and a>b>0, then which of the following is (are) positive? [#permalink]

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New post 14 Jul 2017, 01:21
MathRevolution wrote:
If x>0, y<0, and a>b>0, then which of the following is (are) positive?

Ⅰ. ax+by Ⅱ. ax-by Ⅲ. by-ax

A. Ⅰonly
B. Ⅱ only
C. Ⅲ only
D.Ⅰ& Ⅱ only
E. Ⅱ & Ⅲ only


\(x > 0 > y\) and \(a>b>0\)

(I). If \(|y| >> {x,a,b}\) then \(ax+by < 0\). For example, \(x=2\), \(a=2\), \(b=1\), \(y=-10^{10}\).

(II). It's clear that \(ax>0\) and \(-by>0\). Hence \(ax-by>0\)

(III). \(by < 0\) and \(-ax < 0\). Hence \(by-ax<0\)

Only (II) is true. Answer B
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If x>0, y<0, and a>b>0, then which of the following is (are) positive? [#permalink]

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New post 15 Jul 2017, 14:29
MathRevolution wrote:
If x>0, y<0, and a>b>0, then which of the following is (are) positive?

Ⅰ. ax+by Ⅱ. ax-by Ⅲ. by-ax

A. Ⅰonly
B. Ⅱ only
C. Ⅲ only
D.Ⅰ& Ⅱ only
E. Ⅱ & Ⅲ only

x, a, and b are positive
y is negative
a > b

Let P be the final product of two terms
Let Q be the final product of the other two terms

I. ax+by

[(pos)(pos)] + [(pos)(neg)] =
[pos] + [neg] =
[P] + [-Q] = P - Q: could be positive or negative

We don't have enough information about values.
If |P| > |Q|, it's positive.
If |P| < |Q|, it's negative.

a = 2, x = 2, b = 1, y = -1 -->
(4) + (-1) => 4 - 1 = 3. Positive.

a = 2, x = 2, b = 1, y = -10 -->
(4) + (-10) => 4 - 10 = -6. Negative. REJECT

II. ax-by

[(pos)(pos)] - [(pos)(neg)] =
[pos] - [neg] => (pos) + (pos)
[P] - [-Q] => P + Q ---> ALWAYS positive

Subtracting a negative means addition, which always increases the value of the number to which it is being added (and the first number here is already positive).

a = 2, x = 2, b = 1, y = -10
(4) - (-10) => 4 + 10 = 14. KEEP

III. by-ax

[(pos)(neg)] - [(pos)(pos)] =
[neg] - [pos] =
(-Q) - (P) => -Q - P = ALWAYS negative

Subtracting a positive number from ANY other number always decreases the other number's value (and here, original number is already negative) .

a = 2, x = 1, b = 1, y = -1
(-1) - (2) => -1 - 2 = -3 Negative. REJECT

Only II must be positive. Answer B
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Re: If x>0, y<0, and a>b>0, then which of the following is (are) positive? [#permalink]

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New post 16 Jul 2017, 18:47
==> You getⅠ. ax+by a=2, b=x=1, y=-10, which is negative. (X)
From Ⅱ. ax-by by<0, you get –by>0 and ax>0, ax-by>0, hence it is always positive. (O)
From Ⅲ. by-ax a=2, b=1, x=1, y=-10, it is negative. (X)

Therefore, the answer is B.
Answer: B
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Re: If x>0, y<0, and a>b>0, then which of the following is (are) positive? [#permalink]

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New post 19 Jul 2017, 07:28
MathRevolution

If x>0, y<0, and a>b>0, then which of the following must be positive?

Ⅰ. ax+by Ⅱ. ax-by Ⅲ. by-ax

A. Ⅰonly
B. Ⅱ only
C. Ⅲ only
D.Ⅰ& Ⅱ only
E. Ⅱ & Ⅲ only

Then the ans will B
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Re: If x>0, y<0, and a>b>0, then which of the following is (are) positive? [#permalink]

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New post 06 Aug 2017, 04:45
MathRevolution wrote:
If x>0, y<0, and a>b>0, then which of the following is (are) positive?

Ⅰ. ax+by Ⅱ. ax-by Ⅲ. by-ax

A. Ⅰ only
B. Ⅱ only
C. Ⅲ only
D.Ⅰ& Ⅱ only
E. Ⅱ & Ⅲ only



a>0,x>0 => ax > 0
b>0,y<0 => by < 0

ax+by = (+ve )+ (-ve) >0 can't say
ax-by = (+ve )- (-ve) > 0 true
by-ax = (-ve)-(+ve) > 0 wrong

So, only II is positive

Ans B.
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Re: If x>0, y<0, and a>b>0, then which of the following is (are) positive? [#permalink]

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New post 11 Jun 2018, 12:57
It can be inferred from the information given that only y is negative.

given the constraints only II can be correct. i.e. Ⅱ. ax-by and -ve multiplied by -ve will yield a positive number. In other options it depends on the value of y, hence we cannot confidently say whether the final result will be positive or negative.
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Re: If x>0, y<0, and a>b>0, then which of the following is (are) positive?   [#permalink] 11 Jun 2018, 12:57
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If x>0, y<0, and a>b>0, then which of the following is (are) positive?

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