Let x and y be two real positive

Nathen Peterson

Nathen Peterson

Answered question

2022-03-21

Let x and y be two real positive integers, such that: x+y+xy=3 prove that x+y2.

Answer & Explanation

Ruben Gibson

Ruben Gibson

Beginner2022-03-22Added 9 answers

3=x+y+xyx+y+(x+y2)2,
which gives (x+y)2+4(x+y)12
or (x+y+2)216
or x+y2.
Ali Price

Ali Price

Beginner2022-03-23Added 6 answers

Step 1
x+y+xy=3
x+y+xy+1=3+1
which gives (1+x)(1+y)=4
where (1+x),(1+y)>0
Step 2
Now applying AM-GM inequality:
(1+x)+(1+y)2(1+x)(1+y)
which gives 2+x+y24
or x+y2.

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