Which of the following inequalities holds true?
(A) \(\sqrt{5} + \sqrt{3} \gt \sqrt{6} + \sqrt{2}\).
(B) If \(a \gt b\) and \(c \lt 0\), then \(\displaystyle\frac{a}{c} \lt \frac{b}{c}\).
(C) \(\displaystyle\frac{1}{x^2} \gt \frac{1}{x} \gt 1\), if \(0 \lt x \lt 1\).
(D) If \(a\) and \(b\) are positive integers and \(\displaystyle\frac{a - b}{6.25} = \frac{4}{2.5}\), then \(b \gt a\).
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → (A), (B) and (C) only
(A) \(\sqrt{5} + \sqrt{3} \gt \sqrt{6} + \sqrt{2}\) — Correct. Both sides are positive, so compare their squares:
\( \displaystyle (\sqrt{5} + \sqrt{3})^2 = 8 + 2\sqrt{15} \)
\( \displaystyle (\sqrt{6} + \sqrt{2})^2 = 8 + 2\sqrt{12} \)
Since \(\sqrt{15} \gt \sqrt{12}\), the left side is larger.
(B) If \(a \gt b\) and \(c \lt 0\), then \(\frac{a}{c} \lt \frac{b}{c}\) — Correct. Dividing by a negative number reverses the inequality. For example, \(4 \gt 2\) but \(\frac{4}{-2} = -2 \lt \frac{2}{-2} = -1\).
(C) \(\frac{1}{x^2} \gt \frac{1}{x} \gt 1\), if \(0 \lt x \lt 1\) — Correct. For \(0 \lt x \lt 1\), \(\frac{1}{x} \gt 1\). Multiplying \(\frac{1}{x} \gt 1\) by the positive number \(\frac{1}{x}\) gives \(\frac{1}{x^2} \gt \frac{1}{x}\).
(D) If a and b are positive integers and \(\frac{a - b}{6.25} = \frac{4}{2.5}\), then \(b \gt a\) — Incorrect.
\( \displaystyle a - b = \frac{6.25 \times 4}{2.5} = 10 \)
So \(a - b = 10 \gt 0\), that is, \(a \gt b\).