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Subtracting 1 from each fraction, we get $$-\frac{1}{x+2}-\frac{1}{x+9}=-\frac{1}{x+3}-\frac{1}{x+8}.$$Then $$\frac{1}{x+2}+\frac{1}{x+9}=\frac{1}{x+3}+\frac{1}{x+8},$$so $$\frac{2x+11}{(x+2)(x+9)}=\frac{2x+11}{(x+3)(x+8)}.$$Multiplying both sides by $(x+2)(x+9)(x+3)(x+8),$ we get $$(2x+11)(x+3)(x+8)=(2x+11)(x+2)(x+9).$$Then $$(2x+11)[(x+3)(x+8)-(x+2)(x+9)]=(2x+11)(6)=0.$$Hence, $x=\boxed{-\frac{11}{2}}.$