\(\sqrt{x^2+16}-\sqrt{x^2+7}=3x-8\)
\(\Leftrightarrow\left(\sqrt{x^2+16}-5\right)+\left(4-\sqrt{x^2+7}\right)=3x-9\)
\(\Leftrightarrow\frac{x^2-9}{\sqrt{x^2+16}+5}+\frac{9-x^2}{\sqrt{x^2+7}+4}=3\left(x-3\right)\)
\(\Leftrightarrow\left(x-3\right)\left(\frac{x+3}{\sqrt{x^2+16}+5}-\frac{x+3}{\sqrt{x^2+7}+4}-3\right)=0\)
\(\Leftrightarrow x=3\)