Which of the following is correct? (i) \(\sqrt{7} + \sqrt{2} > \sqrt{6} + \sqrt{3}\)
(ii) \(\sqrt{7} + \sqrt{2} < \sqrt{6} + \sqrt{3}\)
(iii) \(\sqrt{7} + \sqrt{2} = \sqrt{6} + \sqrt{3}\)
(ii)
Step 1 – square both expressions to compare:
\((\sqrt{7} + \sqrt{2})^2 = 7 + 2 + 2\sqrt{14} = 9 + 2\sqrt{14}\)
\((\sqrt{6} + \sqrt{3})^2 = 6 + 3 + 2\sqrt{18} = 9 + 2\sqrt{18}\)
Step 2 – compare:
Since \(\sqrt{18} > \sqrt{14}\) (because \(18 > 14\)), we get \(9 + 2\sqrt{18} > 9 + 2\sqrt{14}\).
Both expressions are positive, so taking square roots preserves the inequality:
\(\sqrt{6} + \sqrt{3} > \sqrt{7} + \sqrt{2}\), i.e. \(\sqrt{7} + \sqrt{2} < \sqrt{6} + \sqrt{3}\).
Hence the correct option is (ii).
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