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Thinking about Algebra

Review question

Given that $a, b > 0$, can we prove that $a/b + b/a \ge 2$?
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Ref: R7354

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Question

Given that \(a > 0\), \(b > 0\), prove that \[\begin{equation*} \frac{a}{b} + \frac{b}{a} \ge 2. \end{equation*}\]
Given that \(x > 0\), \(y > 0\), \(z > 0\) and that \(x + y + z = 3\), prove that \[\begin{equation*} \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \ge 3. \end{equation*}\]