Is |f(x) + C1| the same thing as |f(x)| + C2, where the C's are just arbitrary constants? In other words, can one take out the constant from an absolute value sign?
The context of this is that I'm trying to take the absolute value of an integral whose antiderivative evaluates to f(x) + C.
Thanks in advance!
Set $f(x)=x,~$ $g(x)=\lvert x+C_1\rvert,~$ $h(x)=\lvert x\rvert+C_2$. Assume $g(x)=h(x)$ everywhere.
Then $g(-C_1)=0$ and $h(-C_1)=\lvert C_1 \rvert+C_2.$ Thus, $\lvert C_1 \rvert=-C_2,$ so $C_2 \leq 0$.
$g(0)=\lvert C_1 \rvert$ and $h(0)=C_2$, so $\lvert C_1 \rvert=C_2$ and $C_2 \geq 0.$ Therefore, $C_2=0$. It also means $C_1=0$ contradicting to $C_1$ being arbitrary integration constant.