Is this inequality satisfied for large values?

Question

I'm trying to check whether this inequality holds for large values of $B$

$$\frac{-AB^2}{B-x_2}+\frac{2y_2}{B}<\frac{-AB^2}{B-x_1}+\frac{2y_1}{B}$$

where $x_2>x_1$ $B>x_2$, $y_2>y_1$ and all other variables are positive. Does this inequality hold for large values of $B?$

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Since the $\frac{-AB^2}{B-x_2}$ denominator is smaller than $\frac{-AB^2}{B-x_1}$ it seems intuitively to hold. However, in the limit (where $B \rightarrow \infty$, the two sides should be equal?). How can you check rigorously that this is satisfied?

Answer 1

Note that

$$\frac{-AB^2}{B-x_2}+\frac{2y_2}{B}<\frac{-AB^2}{B-x_1}+\frac{2y_1}{B} \iff \frac{-AB}{1-\frac{x_2}B}+\frac{2y_2}{B}<\frac{-AB}{1-\frac{x_1}B}+\frac{2y_1}{B}$$

$$ (-AB)\left(1+\frac{x_2}B\right)+\frac{2y_2}{B}<(-AB)\left(1+\frac{x_1}B\right)+\frac{2y_1}{B}$$

$$ -AB-Ax_2+\frac{2y_2}{B}<-AB-Ax_1+\frac{2y_1}{B}$$

$$\frac1B(2y_2-2y_1)< A(x_2-x_1)\iff B>\frac{2y_2-2y_1}{A(x_2-x_1)}$$