- #1
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The question was to prove
[tex]\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}>1, x>1[/tex]
And I had two choices to go about this, I could have manipulated the expression
[tex]\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}[/tex]
by multiplying numerator and denominator by its conjugate, squaring, manipulating etc. and getting an obvious result that proves it is more than 1, but instead I went about it a quicker way which before today I thought was logically sound.I started with the assumption that it was true, and would manipulate it from there.
Squaring both sides:
[tex]x+\sqrt{x}+x-\sqrt{x}-2\sqrt{x^2-x}>1[/tex]
Rearranging:
[tex]2x-1>2\sqrt{x^2-x}[/tex]
Squaring, 2x-1>1 since x>1:
[tex]4x^2-4x+1>4x^2-4x[/tex]
[tex]1>0[/tex]
Thus since this result is true, the original statement must have been true.
I ended up getting 1/4 marks for this, and my teacher's reasoning was that it's because I started with the assumption that it was true, and any false statement can lead to a true statement. We threw counter-arguments back and forth, and after asking for another example where this happens, she gave me "if the moon is made of cheese, then cows aren't purple". Honestly, I don't get this woman.
After giving my teacher's argument about false statements leading to truth statements further thought, I admit that problems would arise if
[tex]\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}<-1[/tex]
since when I square both sides, it would lead to a truth statement. I should have proven that the original statement was at least more than -1, but anyway, I'd like to hear from you guys on what you think about my proof, her argument, and where I could improve as I honestly bear more weighting on your word than my teacher's.
[tex]\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}>1, x>1[/tex]
And I had two choices to go about this, I could have manipulated the expression
[tex]\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}[/tex]
by multiplying numerator and denominator by its conjugate, squaring, manipulating etc. and getting an obvious result that proves it is more than 1, but instead I went about it a quicker way which before today I thought was logically sound.I started with the assumption that it was true, and would manipulate it from there.
Squaring both sides:
[tex]x+\sqrt{x}+x-\sqrt{x}-2\sqrt{x^2-x}>1[/tex]
Rearranging:
[tex]2x-1>2\sqrt{x^2-x}[/tex]
Squaring, 2x-1>1 since x>1:
[tex]4x^2-4x+1>4x^2-4x[/tex]
[tex]1>0[/tex]
Thus since this result is true, the original statement must have been true.
I ended up getting 1/4 marks for this, and my teacher's reasoning was that it's because I started with the assumption that it was true, and any false statement can lead to a true statement. We threw counter-arguments back and forth, and after asking for another example where this happens, she gave me "if the moon is made of cheese, then cows aren't purple". Honestly, I don't get this woman.
After giving my teacher's argument about false statements leading to truth statements further thought, I admit that problems would arise if
[tex]\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}<-1[/tex]
since when I square both sides, it would lead to a truth statement. I should have proven that the original statement was at least more than -1, but anyway, I'd like to hear from you guys on what you think about my proof, her argument, and where I could improve as I honestly bear more weighting on your word than my teacher's.