For the resistor circuit shown in the picture, what must be the missing resistance to give a total resistance of 252 ohms?
Your answer is right.  To help gain some intuition ...

Assume that the X is an open circuit, ie. that the series resistance is 22+ 180+ 75 = 277.  that's just a little bit more than the desired result.

Assume that X is a short, then that parallel network is 16//180 = 14.7 (// here means in parallel). the total resistance is 22+14.7+75 = 111.7.

These should tell you that the X value should be large-ish.

Plugging your solution back in:

$$ 22 + \dfrac{1}{\dfrac{1}{1100+16}+\dfrac{1}{180}} + 75 = R_{Total} = 252$$