The question says that $x > 0$ and then we have to prove that $(x + 1)^{1/2} < 1 + (1/2)x$. I tried this question and proved that $(x + 1) < (1 + (1/2)x)^2$ but after this I am not able to proceed because I can't do square root in both sides because this will not be true for fractions and decimals.
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It's not clear what's behind that division sign. Is it $0.5x$ or is it $\frac{1}{2x}$ ? – imranfat Feb 13 '14 at 16:56
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You can solve that using calculus too. See the sign of the derivative of $1+ \frac{x}{2}-\sqrt{x+1}$ at $x>0$. – Sawarnik Feb 13 '14 at 17:20
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Applying the square root to both sides of the inequality is valid here because both sides are positive and the square root of a bigger number is bigger than the square root of a smaller number. Fractions and decimals are just numbers, and their relative sizes behave the same way as those of other numbers. So, apply the square root and you're done.
Bulberage
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