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# If x and y are positive, is x^(1/2) + y^(1/2) ?

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Joined: 27 Oct 2016
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If x and y are positive, is x^(1/2) + y^(1/2) ? [#permalink]

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06 Nov 2017, 05:00
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38% (01:07) correct 62% (01:52) wrong based on 53 sessions

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If x and y are positive, is $$\sqrt{x}+ \sqrt{y} > 1$$ ?

(1) $$\sqrt{x+y}>1$$

(2) $$x > y > \frac{1}{4}$$
[Reveal] Spoiler: OA

Kudos [?]: 6 [0], given: 43

Math Expert
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Re: If x and y are positive, is x^(1/2) + y^(1/2) ? [#permalink]

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06 Nov 2017, 05:37
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If x and y are positive, is $$\sqrt{x}+ \sqrt{y} > 1$$ ?

Is $$\sqrt{x}+ \sqrt{y} > 1$$ ?
Square: is $$x+ 2\sqrt{xy}+y > 1$$ ?

(1) $$\sqrt{x+y}>1$$. Square: x + y > 1. Since x and y are positive, then $$2\sqrt{xy}>0$$, and thus $$x+ 2\sqrt{xy}+y=(x+y)+2\sqrt{xy}=(number \ more \ than \ 1)+(positive \ value)>1$$. Sufficient.

(2) $$x > y > \frac{1}{4}$$. Since both x and y are greater than 1/4, then both $$\sqrt{x}$$ and $$\sqrt{y}$$ are greater than 1/2. Hence, $$\sqrt{x}+ \sqrt{y} > 1$$. Sufficient.

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Kudos [?]: 135380 [0], given: 12691

Math Expert
Joined: 02 Aug 2009
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Re: If x and y are positive, is x^(1/2) + y^(1/2) ? [#permalink]

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06 Nov 2017, 05:38
leshwarnag wrote:
If x and y are positive, is $$\sqrt{x}+ \sqrt{y} > 1$$ ?

(1) $$\sqrt{x+y}>1$$

(2) $$x > y > \frac{1}{4}$$

hi...

$$\sqrt{x}+ \sqrt{y} > 1$$
square both sides....
$$(\sqrt{x}+ \sqrt{y})^2 > 1^2.............x+y+2\sqrt{xy}>1$$

lets see statements

(1) $$\sqrt{x+y}>1$$
square both sides..
$$x+y>1$$...
if $$x+y >1$$, when you add another positive value $$2\sqrt{xy}$$, it will surely be >1
so $$x+y+2\sqrt{xy}>1$$
suff

(2) $$x > y > \frac{1}{4}$$
lets take both as 1/4
so $$\sqrt{x}+ \sqrt{y} = \sqrt{1/4}+\sqrt{1/4}=1/2+1/2=1$$
so the equation will be >1
suff

D
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Absolute modulus :http://gmatclub.com/forum/absolute-modulus-a-better-understanding-210849.html#p1622372
Combination of similar and dissimilar things : http://gmatclub.com/forum/topic215915.html

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If x and y are positive, is x^(1/2) + y^(1/2) ? [#permalink]

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06 Nov 2017, 05:42
Square root of a positive fraction is greater than itself

1. $$\sqrt{x+y} > 1$$
For the above statement to be true: x+y > 1
You can check with any combination of x and 1-x
0.5, 0.5 : $$\sqrt{0.5} + \sqrt{0.5} => 1.4$$
0.1, 0.9 : $$\sqrt{0.1} + \sqrt{0.9} => 0.31 + 0.94 => 1.25$$
0.01, 0.99 : $$\sqrt{0.01} + \sqrt{0.99} => 0.1 + 0.995 => 1.095$$
It would always be greater than 1
Yes

2. $$x > y > \frac{1}{4}$$

$$\sqrt{1/4} = 0.5$$
So $$\sqrt{x} > \sqrt{y} > 0.5$$
$$\sqrt{x} + \sqrt{y} > 1$$
YES

D.
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If x and y are positive, is x^(1/2) + y^(1/2) ?   [#permalink] 06 Nov 2017, 05:42
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