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- #1

- Feb 14, 2012

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Solve the inequality $\sqrt{4x^2-8x+5}+\sqrt{3x^2+12x+16}\ge 6\sqrt{x}-x-6$.

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- Thread starter
- Admin
- #1

- Feb 14, 2012

- 3,894

-----

Solve the inequality $\sqrt{4x^2-8x+5}+\sqrt{3x^2+12x+16}\ge 6\sqrt{x}-x-6$.

-----

Remember to read the POTW submission guidelines to find out how to submit your answers!

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- #2

- Feb 14, 2012

- 3,894

However, I will give the community another week's time to take another stab at the problem. I am looking forward to receiving submissions from the members!

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- #3

- Feb 14, 2012

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Note that we need $x\ge 0$ in order for the right hand side of the inequality to be defined. Moreover, for all non-negative real numbers $x$, we have

$\begin{align*}\sqrt{4x^2-8x+5}+\sqrt{3x^2+12x+16}&=\sqrt{4(x-1)^2+1}+\sqrt{3(x+2)^2+4}\\& \ge 1+2 \\&=3\end{align*}$

On the other hand, $6\sqrt{x}-x-6=-(\sqrt{x}-3)^2+3\le 3$. This completes the proof.

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