Which of the following inequalities holds true?

(A) $\sqrt{5} + \sqrt{3} > \sqrt{6} + \sqrt{2}$.

(B) If $a > b$ and $c < 0$, then $\frac{a}{c} < \frac{b}{c}$.

(C) $\frac{1}{x^2} > \frac{1}{x} > 1$, if $0 < x < 1$.

(D) If $a$ and $b$ are positive integers and $\frac{a - b}{6.25} = \frac{4}{2.5}$, then $b > a$.

Choose the correct answer from the options given below:

Statement (A): $\sqrt{5} + \sqrt{3} > \sqrt{6} + \sqrt{2}$

Both sides are positive, so squaring preserves the inequality direction.

Left side squared:

$(\sqrt{5} + \sqrt{3})^2$

$= 5 + 2\sqrt{15} + 3$

$= 8 + 2\sqrt{15}$

Right side squared:

$(\sqrt{6} + \sqrt{2})^2$

$= 6 + 2\sqrt{12} + 2$

$= 8 + 2\sqrt{12}$

Comparing $8 + 2\sqrt{15}$ vs $8 + 2\sqrt{12}$ reduces to comparing $\sqrt{15}$ vs $\sqrt{12}$.

Since $15 > 12$, we have $\sqrt{15} > \sqrt{12}$.

Statement (A) is TRUE.

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Statement (B): If $a > b$ and $c < 0$, then $\frac{a}{c} < \frac{b}{c}$

When dividing an inequality by a negative number, the inequality sign flips.

Starting with: $a > b$

Dividing both sides by $c$ (where $c < 0$):

$\frac{a}{c} < \frac{b}{c}$

Example: If $5 > 3$ and $c = -2$:

$\frac{5}{-2} = -2.5$

$\frac{3}{-2} = -1.5$

$-2.5 < -1.5$

Statement (B) is TRUE.

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Statement (C): $\frac{1}{x^2} > \frac{1}{x} > 1$, if $0 < x < 1$

Testing with $x = 0.5$:

$\frac{1}{x} = \frac{1}{0.5} = 2$

$\frac{1}{x^2} = \frac{1}{0.25} = 4$

This gives $4 > 2 > 1$.

For $0 < x < 1$:

Since $x < 1$, we have $\frac{1}{x} > 1$.

Since $0 < x < 1$, multiplying by $x$ gives $x^2 < x$.

Taking reciprocals of positive numbers reverses the inequality: $\frac{1}{x^2} > \frac{1}{x}$.

Statement (C) is TRUE.

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Statement (D): If $\frac{a - b}{6.25} = \frac{4}{2.5}$, then $b > a$

Simplifying the right side:

$\frac{4}{2.5} = \frac{4}{\frac{5}{2}}$

$= 4 \times \frac{2}{5}$

$= \frac{8}{5}$

$= 1.6$

Solving for the relationship between $a$ and $b$:

$\frac{a - b}{6.25} = 1.6$

$a - b = 1.6 \times 6.25$

$a - b = 10$

$a = b + 10$

This means $a > b$, NOT $b > a$.

Statement (D) is FALSE.

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Statements (A), (B), and (C) are TRUE.