# Prove that it is smaller than 6

• August 28th 2011, 07:06 AM
MichaelLight
Prove that it is smaller than 6
Prove that Attachment 22141.

How do i prove this? thank you...
• August 28th 2011, 07:10 AM
Siron
Re: Prove that it is smaller than 6
A possibility is to take the square of both sides.
• August 28th 2011, 07:12 AM
alexmahone
Re: Prove that it is smaller than 6
Quote:

Originally Posted by MichaelLight
Prove that Attachment 22141.

How do i prove this? thank you...

$\sqrt{10}+2\sqrt{2}<6$

$\Leftrightarrow 10+8+8\sqrt{5}<36$ (Squaring both sides)

$\Leftrightarrow 8\sqrt{5}<18$

$\Leftrightarrow 320<324$ (Squaring both sides)

which is true.
• August 28th 2011, 07:13 AM
CaptainBlack
Re: Prove that it is smaller than 6
Quote:

Originally Posted by MichaelLight
Prove that Attachment 22141.

How do i prove this? thank you...

Suppose otherwise, then by assumption:

$\sqrt{10}+2\sqrt{2}\ge 6$

Now square both sides and follow the consequences to get a contradiction.

CB
• August 28th 2011, 02:11 PM
HallsofIvy
Re: Prove that it is smaller than 6
Quote:

Originally Posted by alexmahone
$\sqrt{10}+2\sqrt{2}<6$

$\Leftrightarrow 10+8+8\sqrt{5}<36$ (Squaring both sides)

$\Leftrightarrow 8\sqrt{5}<18$

$\Leftrightarrow 320<324$ (Squaring both sides)

which is true.

Normally, one would NOT start a proof by assuming what you want to prove, as alexmahone does here. However, this is a perfectly valid "synthetic proof". Every step is invertible. A "standard" proof would be given by going from "320< 324" upward.