Inequalities (Foundations)

Inequalities: core idea

• 1. $x^2\ge0$ always? $1/0$

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• 2. AM-GM: for $a=b=2$, $(a+b)/2$ vs $\sqrt{ab}$: equal? $1/0$

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• 3. Solve $|x|<1$: number of integers $x$?

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• 4. Solve $x^2<4$: greatest integer $x$?

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• 5. Min of $(x-1)^2$?

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• 6. $a^2+b^2\ge2ab$: for $a=3,b=1$ left−right?

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• 7. Solve $2x\le10$: max integer $x$?

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• 8. Cauchy: $(1^2+2^2)(3^2+4^2)\ge(1\cdot3+2\cdot4)^2$: right side?

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• 9. Solve $-x<3$: least integer $x$?

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