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# Why this is illegal?

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Manager
Joined: 12 Dec 2012
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18 Jan 2013, 14:40
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Could someone please explain me why this is illegal?:

$$\sqrt{x}+\sqrt{y}>0$$

$$\sqrt{x}>-\sqrt{y}$$

$$(\sqrt{x})^2>(-\sqrt{y})^2$$

$$x>y$$

While this is legal:

$$\sqrt{x}-\sqrt{y}>0$$

$$\sqrt{x}>\sqrt{y}$$

$$(\sqrt{x})^2>(\sqrt{y})^2$$

$$x>y$$

Thank you.
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Re: Why this is illegal? [#permalink]

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18 Jan 2013, 15:03
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HumptyDumpty wrote:
Could someone please explain me why this is illegal?:

$$\sqrt{x}+\sqrt{y}>0$$

$$\sqrt{x}>-\sqrt{y}$$

$$(\sqrt{x})^2>(-\sqrt{y})^2$$

$$x>y$$

While this is legal:

$$\sqrt{x}-\sqrt{y}>0$$

$$\sqrt{x}>\sqrt{y}$$

$$(\sqrt{x})^2>(\sqrt{y})^2$$

$$x>y$$

Thank you.

Writing $$x>y$$ from $$\sqrt{x}+\sqrt{y}>0$$ is not right. We have that the sum of two non-negative values ($$\sqrt{x}$$ and $$\sqrt{y}$$) is greater than zero. How can we know that $$x>y$$ from that? With the same logic you could get that $$y>x$$. Right?

Algebraic explanation: we can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality).
For example:
$$2<4$$ --> we can square both sides and write: $$2^2<4^2$$;
$$0\leq{x}<{y}$$ --> we can square both sides and write: $$x^2<y^2$$;
But consider the case when one side is negative: $$-2<1$$ --> if we square we get $$4<1$$, which is not right. So, squaring an inequality where one side is negative won't always give the correct result.

In the first case $$-\sqrt{y}\leq{0}$$, thus we cannot apply squaring. While in the second case both parts of the inequality are non-negatve, thus we can safely square.

GENERAL RULE:
A. We can raise both parts of an inequality to an even power if we know that both parts of an inequality are non-negative (the same for taking an even root of both sides of an inequality).
For example:
$$2<4$$ --> we can square both sides and write: $$2^2<4^2$$;
$$0\leq{x}<{y}$$ --> we can square both sides and write: $$x^2<y^2$$;

But if either of side is negative then raising to even power doesn't always work.
For example: $$1>-2$$ if we square we'll get $$1>4$$ which is not right. So if given that $$x>y$$ then we can not square both sides and write $$x^2>y^2$$ if we are not certain that both $$x$$ and $$y$$ are non-negative.

B. We can always raise both parts of an inequality to an odd power (the same for taking an odd root of both sides of an inequality).
For example:
$$-2<-1$$ --> we can raise both sides to third power and write: $$-2^3=-8<-1=-1^3$$ or $$-5<1$$ --> $$-5^2=-125<1=1^3$$;
$$x<y$$ --> we can raise both sides to third power and write: $$x^3<y^3$$.

Hope it helps.

P.S. Can you please post the question from which you took that example? Thank you.
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Re: Why this is illegal? [#permalink]

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18 Jan 2013, 16:09
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It helped a lot, thanks! The problem posted in DS Section:

a-x-y-and-b-x-y-if-a-2-b-2-what-is-the-value-of-y-146004.html
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Re: Why this is illegal? [#permalink]

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18 Jan 2013, 16:20
HumptyDumpty wrote:
It helped a lot, thanks! The problem posted in DS Section:

a-x-y-and-b-x-y-if-a-2-b-2-what-is-the-value-of-y-146004.html

Just posted a solution: a-x-y-and-b-x-y-if-a-2-b-2-what-is-the-value-of-y-146004.html#p1170874

By the way, we can solve this question without squaring inequalities.
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Re: Why this is illegal? [#permalink]

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18 Jan 2013, 16:50
Bunuel wrote:

Just posted a solution: a-x-y-and-b-x-y-if-a-2-b-2-what-is-the-value-of-y-146004.html#p1170874

By the way, we can solve this question without squaring inequalities.

Yes that I know, but I had that gap which you've just dismissed .
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Re: Why this is illegal? [#permalink]

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31 Jan 2013, 01:44
sqrt x + sqrt y = 0 fine

sqrt can only be positive

sum of these 2 positive numbers can be 0 only if each is equal to zero so only one possibility.

sqrt x = - sqrt y LHS positive, RHS negative.... ????? again, each side should be 0

Now when u r trying to square each side,

u r multiply with a positive quantity on LHS and negative quantity on RHS

Inequalities should be either multiplied by a positive number or a negative number on each side, with sign changing in the latter case.

Thanks.

KUDOS if u appreciate
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Re: Why this is illegal? [#permalink]

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20 Nov 2014, 06:05
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Re: Why this is illegal?   [#permalink] 20 Nov 2014, 06:05
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