Solving the Mystery of sinh(-3) ≥ -3 or sinh(-3) ≤ -3?

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The discussion revolves around the inequalities involving the hyperbolic sine function, specifically sinh(-3). The user initially concludes that sinh(-3) ≥ -3 based on the property exp(y) ≥ 1 + y. However, upon further manipulation, they derive sinh(-3) ≤ -3, leading to confusion about which inequality is correct. The key takeaway is the importance of careful handling of inequalities, especially when multiplying by negative numbers, which can reverse the direction of the inequality. The thread highlights a common pitfall in mathematical reasoning related to inequalities.
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I'm given the fact that exp(y) ≥ 1 + y for all y

I will use this for sinh. Since sinh = (exp(y) - exp(-y))/2 we have for y=3 as example:

sinh(3) ≥ 3 by subtracting the two inequalities

Instead for sinh(-3) I get:

sinh(-3) ≥ -3 (1)

But something is wrong here: Because if I take the first inequality sinh(3) ≥ 3 and multiply it by (-1) and switch around the inequality sinh and using the fact that
-sinh(3) = sinh(-3) I get:

sinh(-3) ≤ -3 (2)

This is clearly weird. Which of (1) and (2) is right and why is other one wrong?
 
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Careful when subtracting inequalities:

5 > 3
4 > 1

subtracting:
1 > 2 ?
 
ahh omg. Thanks LOL
 

Thread 'Greatest possible value of a constant in polynomial'

Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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