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Inequalities

From the basics to AM-GM and Cauchy-Schwarz.

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What is the largest real number \(x\) that satisfies \(2x+3\geq 4x+9\)?

If the product of 5 positive real numbers is 3125, what is the minimum value of their sum?

Let \(a,b,c,d\) be positive real numbers such that \[a+b+c + d = 1.\] What is the minimum value of \[\frac{1}{a} + \frac{4}{b} + \frac{9}{c} + \frac{16}{d}?\]

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