If (x + 5) is the mean proportion between x + 2 and x + 9, find the value of x.

Since, (x + 5) is the mean proportion between x + 2 and x + 9,

$\therefore \dfrac{x + 2}{x + 5} = \dfrac{x + 5}{x + 9} \\[0.5em] \Rightarrow (x + 5)^2 = (x + 2)(x + 9) \\[0.5em] \Rightarrow (x^2 + 25 + 10x) = (x^2 + 9x + 2x + 18) \\[0.5em] \Rightarrow x^2 - x^2 + 10x - 11x + 25 - 18 = 0 \\[0.5em] \Rightarrow -x + 7 = 0 \\[0.5em] \Rightarrow x = 7.$

Hence, the value of x is 7.