Could I please have help solving this inequality. I have tried but haven't made meaningful progress. If x,y and z are real numbers prove that: 1/x + 1/y +1/z >= (9)/(x+y+z)

Ethen Blackwell

Ethen Blackwell

Answered question

2022-07-17

Inequality Problem 1 x + 1 y + 1 z 9 x + y + z
Could I please have help solving this inequality. I have tried but haven't made meaningful progress.
If x,y and z are real numbers prove that:
1 x + 1 y + 1 z 9 x + y + z

Answer & Explanation

fairymischiefv9

fairymischiefv9

Beginner2022-07-18Added 11 answers

Multiply your inequality by x y z ( x + y + z ), you get :
y 2 z + y z 2 + x 2 z + x z 2 + x 2 y + x y 2 6 x y z
Now redivide by x y z (maybe not a good idea to multiply initially :-) :
y x + z x + x y + z y + x z + y z 6
As it is well know that for all u > 0, u + 1 u 2, you get your inequality for x , y , z > 0
Now for the case x , y , z non null reals, I don't know...
Livia Cardenas

Livia Cardenas

Beginner2022-07-19Added 5 answers

Obviously we must assume that the numbers are positive, as otherwise the inequality doesn't always hold. Nevertheless you can prove this inequality by using Cauchy-Schwartz Inequality. We have:
( 1 x + 1 y + 1 z ) ( x + y + z ) ( x x + y y + z z ) 2 = 9
Now dividing by ( x + y + z ) we get the wanted inequality.

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