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.

_________________

New to the Math Forum?

Please read this: All You Need for Quant | PLEASE READ AND FOLLOW: 12 Rules for Posting!!!

Resources:

GMAT Math Book | Triangles | Polygons | Coordinate Geometry | Factorials | Circles | Number Theory | Remainders; 8. Overlapping Sets | PDF of Math Book; 10. Remainders | GMAT Prep Software Analysis | SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) | Tricky questions from previous years.

Collection of Questions:

PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.

What are GMAT Club Tests?

Extra-hard Quant Tests with Brilliant Analytics