Circles with centers of $(2,2)$ and $(17,10)$ are both tangent to the $x$-axis. What is the distance between the closest points of the two circles?
The radius of the first circle is 2, and the radius of the second circle is 10.  The distances between the centers of the circles is $\sqrt{(17 - 2)^2 + (10 - 2)^2} = 17,$ so the distance between the two closest points of the two circle is $17 - 2 - 10 = \boxed{5}.$

[asy]
unitsize(0.3 cm);

draw((2,2)--(2,0),dashed);
draw((17,10)--(17,0),dashed);
draw((-1,0)--(28,0));
draw((0,-1)--(0,20));
draw(Circle((2,2),2));
draw(Circle((17,10),10));
draw((2,2)--(17,10));

label("$2$", (2,1), E);
label("$10$", (17,5), E);

dot("$(2,2)$", (2,2), NW);
dot("$(17,10)$", (17,10), NE);
[/asy]