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# Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1

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Joined: 23 May 2018
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Location: Pakistan
Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1  [#permalink]

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30 Sep 2018, 02:48
Manbehindthecurtain wrote:
Are x and y both positive?

(1) $$2x - 2y = 1$$

(2) $$\frac{x}{y} > 1$$

(1) $$2x - 2y = 1$$
$$2(x - y) = 1$$
$$x - y = \frac{1}{2}$$
$$x = y + \frac{1}{2}$$

INSUFFICIENT

(2) $$\frac{x}{y} > 1$$
Only thing we know is that x and y are either both positive or both negative.

INSUFFICIENT

Together

$$\frac{x}{y} > 1$$
$$1 + \frac{y}{2} > 1$$
$$\frac{y}{2} > 0$$

Thus, y is greater than 0 and as x is greater than y, we know that both are positive.

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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1  [#permalink]

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29 Nov 2018, 23:23
1
Manbehindthecurtain wrote:
Are x and y both positive?

(1) $$2x - 2y = 1$$

(2) $$\frac{x}{y} > 1$$

Quote:
If we are given a statement x/y>1, and we are asked is x and y both positive: is my below mentioned approach correct or is there some gap in my understanding, please help!.

x/y >1
Multiplying above equation by by y^2 both side-------------------------->we can write it as xy>y^2
Therefore,
y(x-y)>0
y(y-x)<0 (sign flips on converting x-y to y-x)
so we have 2 points on number line now 0 and x. hence, y lies between 0 and x --->(0<y<x)
this gives us both x and y are more than zero.
hence, x n y are both positive.

Interesting manipulations! But here is the problem: y lies between 0 and x --->(0<y<x)

y lies between 0 and x does not translate to 0 < y < x.
What if x is negative? Then x < y < 0
Say if x is -3, y will be between -3 and 0.

SO both could be positive BUT both could be negative too.
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1  [#permalink]

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02 Feb 2019, 08:20
Top Contributor
Manbehindthecurtain wrote:
Are x and y both positive?

(1) $$2x - 2y = 1$$

(2) $$\frac{x}{y} > 1$$

Target question: Are x and y both positive?

Statement 1: 2x - 2y = 1
There are several pairs of numbers that satisfy this condition. Here are two:
Case a: x = 1 and y = 0.5, in which case x and y are both positive
Case b: x = -0.5 and y = -1, in which case x and y are not both positive
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: x/y > 1
This tells us that x/y is positive. This means that either x and y are both positive or x and y are both negative. Here are two possible cases:
Case a: x = 4 and y = 2, in which case x and y are both positive
Case b: x = -4 and y = -2, in which case x and y are not both positive
Since we cannot answer the target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2
Statement 1 tells us that 2x - 2y = 1.
Divide both sides by 2 to get: x - y = 1/2
Solve for x to get x = y + 1/2

Now take the statement 2 inequality (x/y > 1) and replace x with y + 1/2 to get:
(y + 1/2)/y > 1
Rewrite as: y/y + (1/2)/y > 1
Simplify: 1 + 1/(2y) > 1
Subtract 1 from both sides: 1/(2y) > 0
If 1/(2y) is positive, then y must be positive.

Statement 2 tells us that either x and y are both positive or x and y are both negative.
Now that we know that y is positive, it must be the case that x and y are both positive
Since we can now answer the target question with certainty, the combined statements are SUFFICIENT

Cheers,
Brent
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Re: Are x and y both positive? (1) 2x - 2y = 1 (2) x/y > 1   [#permalink] 02 Feb 2019, 08:20

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