Solve the following equation for $x$:
$$\frac{1}{x+1}+\frac{2}{x+5}=\frac{1}{2}$$

Combine the fractional terms on the left-hand side using a common denominator:
$$\frac{(x+5)+2(x+1)}{(x+1)(x+5)}=\frac{1}{2}$$

$$\frac{x+5+2x+2}{x^2+6x+5}=\frac{1}{2}$$

$$\frac{3x+7}{x^2+6x+5}=\frac{1}{2}$$

Cross-multiply to get rid of the denominators:

$$2(3x+7)=1(x^2+6x+5)$$

$$6x+14=x^2+6x+5$$

Subtract $6x$ from both sides:

$$14=x^2+5$$

$$x^2=14-5=9$$

$$x=\pm 3$$