Đáp án:
\[\left[ \begin{array}{l}
x = - \frac{1}{2}\\
x = - 4
\end{array} \right.\]
Giải thích các bước giải:
ĐKXĐ: \(\left\{ \begin{array}{l}
x \ne - 2\\
x \ne - 3\\
x \ne - 5\\
x \ne - 6
\end{array} \right.\)
Ta có;
\[\begin{array}{l}
\frac{{x - 1}}{{x + 2}} - \frac{{x - 2}}{{x + 3}} - \frac{{x - 4}}{{x + 5}} + \frac{{x - 5}}{{x + 6}} = 0\\
\Leftrightarrow \frac{{\left( {x - 1} \right)\left( {x + 3} \right) - \left( {x - 2} \right)\left( {x + 2} \right)}}{{\left( {x + 2} \right)\left( {x + 3} \right)}} - \frac{{\left( {x - 4} \right)\left( {x + 6} \right) - \left( {x - 5} \right)\left( {x + 5} \right)}}{{\left( {x + 5} \right)\left( {x + 6} \right)}} = 0\\
\Leftrightarrow \frac{{{x^2} + 2x - 3 - {x^2} + 4}}{{{x^2} + 5x + 6}} - \frac{{{x^2} + 2x - 24 - {x^2} + 25}}{{{x^2} + 11x + 30}} = 0\\
\Leftrightarrow \frac{{2x + 1}}{{{x^2} + 5x + 6}} - \frac{{2x + 1}}{{{x^2} + 11x + 30}} = 0\\
\Leftrightarrow \left[ \begin{array}{l}
2x + 1 = 0\\
{x^2} + 5x + 6 = {x^2} + 11x + 30
\end{array} \right. \Leftrightarrow \left[ \begin{array}{l}
x = - \frac{1}{2}\\
x = - 4
\end{array} \right.
\end{array}\]
