Đáp án:
\(x = 15\)
Giải thích các bước giải:
\[\begin{array}{l}
\frac{3}{{\left( {x + 2} \right)\left( {x + 5} \right)}} + \frac{5}{{\left( {x + 5} \right)\left( {x + 10} \right)}} + \frac{7}{{\left( {x + 10} \right)\left( {x + 17} \right)}} = \frac{x}{{\left( {x + 2} \right)\left( {x + 17} \right)}}\\
DK:\,\,\,\left\{ \begin{array}{l}
x \ne - 2\\
x \ne - 5\\
x \ne - 10\\
x \ne - 17
\end{array} \right.\\
pt \Leftrightarrow \frac{{x + 5 - \left( {x + 2} \right)}}{{\left( {x + 2} \right)\left( {x + 5} \right)}} + \frac{{x + 10 - \left( {x + 5} \right)}}{{\left( {x + 5} \right)\left( {x + 10} \right)}} + \frac{{x + 17 - \left( {x + 10} \right)}}{{\left( {x + 10} \right)\left( {x + 17} \right)}} = \frac{x}{{\left( {x + 2} \right)\left( {x + 17} \right)}}\\
\Leftrightarrow \frac{1}{{x + 2}} - \frac{1}{{x + 5}} + \frac{1}{{x + 5}} - \frac{1}{{x + 10}} + \frac{1}{{x + 10}} - \frac{1}{{x + 17}} = \frac{x}{{\left( {x + 2} \right)\left( {x + 17} \right)}}\\
\Leftrightarrow \frac{1}{{x + 2}} - \frac{1}{{x + 17}} = \frac{x}{{\left( {x + 2} \right)\left( {x + 17} \right)}}\\
\Leftrightarrow x + 17 - x - 2 = x\\
\Leftrightarrow x = 15\,\,\,\left( {tm} \right).\\
Vay\,\,\,x = 15.
\end{array}\]
