6. For any two points $P_{1}\left(x_{1}, y_{1}\right), P_{2}\left(x_{2}, y_{2}\right)$ in a Cartesian coordinate system, the expression $\left|x_{1}-x_{2}\right|+\left|y_{1}-y_{2}\right|$ is called the Manhattan distance between $P_{1}$ and $P_{2}$, denoted as $d\left(P_{1}, P_{2}\right)$. If $P_{0}\left(x_{0}, y_{0}\right)$ is a fixed point and $Q(x, y)$ is a moving point on the line $y=k x + b$, the minimum value of $d\left(P_{0}, Q\right)$ is called the Manhattan distance from point $P_{0}$ to the line $y=k x+b$. What is the Manhattan distance $d$ from point $M(1,1)$ to the line $y=2 x+5$?
(A) 3
(B) $\frac{7}{2}$
(C) 4
(D) $\frac{9}{2}$
6. A.

From the condition, we know that the required perpendicular distance $d$ is the minimum value of
$$
S=|x-1|+|2 x+5-1|
$$

which is
$$
\begin{array}{l}
\text { Also, } S=|x-(-2)|+|x-1|+|x-(-2)| \\
\geqslant 3+|x-(-2)| \geqslant 3,
\end{array}
$$

with equality holding if and only if $x=-2$.
Therefore, $d=3$.