#Inequality proof

20 messages · Page 1 of 1 (latest)

small lava
eternal cobaltBOT
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mortal socket
# small lava

You want to show
[\qty( a + \frac{1}{a + 1} ) + \qty( b + \frac{1}{b + 1} ) \geq 2]

paper apexBOT
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invariance.

757741186432630884.png
mortal socket
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because the (a) and (b) parts are independent of each other, this is the same thing as showing
[\qty(x + \frac{1}{x + 1}) \geq 1]
when (x \geq 0).

paper apexBOT
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invariance.

757741186432630884.png
small lava
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Would this be the right way to prove it?

image.png
mortal socket
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note that you cam multiply by x + 1 without changing the sign because x + 1 ≥ 0

small lava
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Oh, so that's why photomath and wolframalpha say that the solution is x > -1?

mortal socket
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yep! your proof works as long as x + 1 is positive, which is when x > -1

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,w plot x + 1/(x + 1)

paper apexBOT
small lava
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So if there was no x >= 0 I should've reviewed x^2 + x + 1 >= 1 and x + 1 >= 1 as well as x^2+x+1 <= 1 and x+1 <= 1?

mortal socket
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if there's no condition on x, the statement isn't always true

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but yeah, if you were trying to find the most general solution, you'd consider both the cases of x + 1 > 0 and x + 1 < 0

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(when you multiplied by x + 1)

small lava
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Thanks!

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