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BPT
$\iff 5x-4+\sqrt{x^{2}+24}-\sqrt{x^{2}+35}>0$
$\iff (5x-5)+(\sqrt{x^{2}+24}-5)-(\sqrt{x^{2}+35}-6)>0$
$\iff 5(x-1)+\dfrac{x^2-1}{\sqrt{x^{2}+24}+5}-\dfrac{x^2-1}{\sqrt{x^{2}+35}+6}>0$
$\iff (x-1)\left (5+\dfrac{x+1}{\sqrt{x^{2}+24}+5}+\dfrac{-x-1}{\sqrt{x^{2}+35}+6} \right )>0$
$\iff (x-1)\underbrace{\left (3+\dfrac{\sqrt{x^{2}+24}+x+6}{\sqrt{x^{2}+24}+5}+\dfrac{\sqrt{x^{2}+35}-x+5}{\sqrt{x^{2}+35}+6} \right )}_{A}>0$
Chú ý rằng
$\sqrt{x^{2}+24}+x+6>\sqrt{x^{2}}+x=|x|+x\ge 0 $
$\sqrt{x^{2}+35}-x+5>\sqrt{x^{2}}-x=|x|-x\ge 0 $
Như vậy $A>0 \iff \boxed{x>1.}$
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