Example 6 In the rectangular coordinate system $x O y$, the moving point $M(x, 0)$ on the $x$-axis has distances $M P$ and $M Q$ to the fixed points $P(5,5)$ and $Q(2,1)$, respectively. Then, when $M P + M Q$ takes the minimum value, the abscissa $x=$ $\qquad$
(2001, National Junior High School Mathematics Competition)
Solution: Make the symmetric point $Q^{\prime}$ of point $Q$ with respect to the $x$-axis, and connect $P Q^{\prime}$ and $M Q^{\prime}$. Then $M P + M Q^{\prime} \geqslant P Q^{\prime}$. Therefore, the minimum value of $M P + M Q$ is $P Q^{\prime}$. At this time, the coordinates of the symmetric point $Q^{\prime}$ of point $Q$ are $(2, -1)$.
Thus, the equation of the line $P Q^{\prime}$ is $y = 2x - 5$.
From this, the x-coordinate of the intersection of $P Q^{\prime}$ with the $x$-axis is $\frac{5}{2}$.