Решите уравнение: \(\dfrac{4x}{x-3} - \dfrac{33}{2x-5} = 5.\)

Решение

\[\dfrac{4x}{x-3} - \dfrac{33}{2x-5} - 5 = 0 \Leftrightarrow\quad \\ \]

\[\Leftrightarrow \dfrac{4x(2x-5) }{(x-3)(2x-5)} - \dfrac{33(x-3)}{(x-3)(2x-5)} - \dfrac{5(x-3)(2x-5)}{(x-3)(2x-5)} = 0 \Leftrightarrow \\\]

\[\Leftrightarrow \dfrac{4x(2x-5) - 33(x-3) - 5(x-3)(2x-5)}{(x-3)(2x-5)} = 0 \Leftrightarrow \\\]

\[\Leftrightarrow \dfrac{8x^2 - 20x - 33x + 99 -5(2x^2 - 5x -6x +15)}{(x-3)(2x-5)} = 0\Leftrightarrow \\\]

\[\Leftrightarrow \dfrac{8x^2 - 53x + 99 - 10x^2 + 55x -75}{(x-3)(2x-5)} = 0\Leftrightarrow \\\]

\[\Leftrightarrow \dfrac{-2x^2 + 2x + 24}{(x-3)(2x-5)} = 0 \Leftrightarrow
\left\{\begin{array}{l}
x^2 - x -12 = 0 \\
x-3 \ne 0 \\
2x-5 \ne 0
\end{array}\right.\Leftrightarrow \]

\[\Leftrightarrow
\left\{\begin{array}{l}
(x - 4)(x + 3) = 0 \\
x \ne 3 \\
x \ne 2,5
\end{array}\right.
\Leftrightarrow
\left[
\begin{array}{l}
x = 4 \\
x = -3
\end{array}\right.
\]

Ответ: \(4; -3\).