<s>[INST] $\frac{3x+1}{x+1}+\frac{3y+1}{y+1}+\frac{3z+1}{z+1} \le \frac{9}{2}$ I'm having trouble proving that for any $x,y,z>0$ such that $x+y+z=1$ the following inequality is true:
$\frac{3x+1}{x+1}+\frac{3y+1}{y+1}+\frac{3z+1}{z+1} \le \frac{9}{2}$
It seems to me that Jensen's inequality could do the trick, but I'm having trouble finding the right function and the right arguments. Any help is appreciated.
 [/INST] Hint:
$$\left(\frac1{x+1}+\frac1{y+1}+\frac1{z+1}\right)(x+1+y+1+z+1) \ge 9$$
And
$$\frac{3x+1}{x+1} = 3 - \frac{2}{x+1}.$$
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