Đáp án:

$3)x=\frac{-5}{6}\\
4)x=0\\
6) x=2\\
7) S=\varnothing $

Giải thích các bước giải:

$3) \frac{3x+2}{2}-\frac{3x+1}{6}=2x+\frac{5}{3}\\
\Leftrightarrow \frac{3(3x+2)}{6}-\frac{3x+1}{6}=\frac{6.2x}{6}+\frac{5.2}{6}\\
\Leftrightarrow 9x+6-3x-1=12x+10\\
\Leftrightarrow 6x-12x=10-5\\
\Leftrightarrow -6x=5\\
\Leftrightarrow x=\frac{-5}{6}\\
4) \frac{x}{2x-6}+\frac{x}{2x+2}=\frac{-2x}{(3-x)(x+1)}\\
ĐK: {\left\{\begin{aligned}3-x \neq 0\\x+1 \neq 0\end{aligned}\right.} \Leftrightarrow {\left\{\begin{aligned}x \neq 3\\ x \neq -1\end{aligned}\right.}\\
\Rightarrow  \frac{x}{2(x-3)}+\frac{x}{2(x+1)}=\frac{2x}{(x-3)(x+1)}\\
\Leftrightarrow \frac{x(x+1)}{2(x-3)(x+1)}+\frac{x(x-3)}{2(x+1)(x-3)}=\frac{2.2x}{2(x-3)(x+1)}\\
\Leftrightarrow x^2+x+x^2-3x=4x\\
\Leftrightarrow 2x^2-2x-4x=0\\
\Leftrightarrow 2x^2-6x=0\\
\Leftrightarrow 2x(x-3)=0\\
\Leftrightarrow {\left[\begin{aligned}x=0\\x-3=0\end{aligned}\right.}\\
\Leftrightarrow {\left[\begin{aligned}x=0\\x=3(loai) \end{aligned}\right.}\\ \Leftrightarrow x=0\\
6) \frac{5x+2}{6}-\frac{8x-1}{3}=\frac{4x+2}{5}-5\\
\Leftrightarrow \frac{5(5x+2)}{30}-\frac{10(8x-1)}{30}=\frac{6(4x+2)}{30}-\frac{30.5}{30}\\
\Leftrightarrow 25x+10-80x+10=24x+12-150\\
\Leftrightarrow -55x+20-24x+138=0\\
\Leftrightarrow -79x=-158\\
\Leftrightarrow x=2\\
7) \frac{x+5}{x^2-5x}-\frac{x-5}{2x^2+10x}=\frac{x+25}{2x^2-50}\\
ĐK: {\left\{\begin{aligned}x\neq 0\\x-5 \neq 0\\x+5 \neq 0\end{aligned}\right.}\Leftrightarrow {\left\{\begin{aligned}x\neq 0\\x \neq 5\\x\neq -5\end{aligned}\right.}\\
\Rightarrow \frac{x+5}{x(x-5)}-\frac{x-5}{2x(x+5)}=\frac{x+25}{2(x^2-25)}\\
\Leftrightarrow \frac{2(x+5)^2}{2x(x-5)(x+5)}-\frac{(x-5)^2}{2x(x+5)(x-5)}=\frac{x(x+25)}{2x(x^2-25)}\\
\Leftrightarrow 2(x^2+10x+25)-(x^2-10x+25)=x^2+25x\\
\Leftrightarrow 2x^2+20x+50-x^2+10x-25-x^2-25x=0\\
\Leftrightarrow 5x+25=0\\
\Leftrightarrow 5x=-25\\
\Leftrightarrow x=-5$  (loại)