(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

*Sorry I could not get the math symbols to work properly so I did it by hand. I hope this isn't too much trouble.

Prove:

| sqrt( x ) - sqrt( y ) | <= | sqrt ( x - y ) |

for x, y >= 0

Hint: Treat the cases x >= y and x <= y separately.

I am new to proofs and we can't use calculus. It's all confusing to me and we've only just begun. The farthest we got was to the Triangle Inequality / the AGM Inequality so I assume that's the most we can do.

2. Relevant equations

I assume whomever is helping me already knows the AGM and Triangle Inequalities since they're the most basic of all the inequalities for proofs.

3. The attempt at a solution

I squared both sides and moved the absolute value signs to the individual variables and then in then in either case, I can remove them.

On the left side I had x + y and just removed that and -2 sqrt(x) sqrt(y) and removed that one too. On the right I had x - y so I write it as that if x > y and -y + x if x < y

Subtract x from both sides and then

y - 2 * sqrt( x ) * sqrt( y ) <= -y

I am pretty lost here. Iff x >= y then the roots would be the same and I could replace sqrt( y ) with sqrt ( x ). I would then add y to both sides and thus

2y - 2x <= 0

2(y-x) <= 0

and that's not true since x >= y and the equality would only hold true in that situation if they were equal, which we know the equality is true if y > x from substitution

PS. Thank you for taking your time to review my question and for whatever help you provide. I appreciate it very much, this question frustrated me for far too long so far.

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Proofs using absolute value with Triangle/AGM Inequality

**Physics Forums | Science Articles, Homework Help, Discussion**