$$\left. \begin{array} { l } { \frac { 1 } { x + 2 } + \frac { 1 } { x + 3 } = \frac { 2 } { x + 9 } } \end{array} \right.$$
$x = -\frac{33}{13} = -2\frac{7}{13} \approx -2.538461538$
$$\left(x+3\right)\left(x+9\right)+\left(x+2\right)\left(x+9\right)=\left(x+2\right)\left(x+3\right)\times 2$$
$$x^{2}+12x+27+\left(x+2\right)\left(x+9\right)=\left(x+2\right)\left(x+3\right)\times 2$$
$$x^{2}+12x+27+x^{2}+11x+18=\left(x+2\right)\left(x+3\right)\times 2$$
$$2x^{2}+12x+27+11x+18=\left(x+2\right)\left(x+3\right)\times 2$$
$$2x^{2}+23x+27+18=\left(x+2\right)\left(x+3\right)\times 2$$
$$2x^{2}+23x+45=\left(x+2\right)\left(x+3\right)\times 2$$
$$2x^{2}+23x+45=\left(x^{2}+5x+6\right)\times 2$$
$$2x^{2}+23x+45=2x^{2}+10x+12$$
$$2x^{2}+23x+45-2x^{2}=10x+12$$
$$23x+45=10x+12$$
$$23x+45-10x=12$$
$$13x+45=12$$
$$13x=12-45$$
$$13x=-33$$
$$x=\frac{-33}{13}$$
$$x=-\frac{33}{13}$$
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