$\text{a)}$  

Zał:

$x\ne 0,2+x\ne 0$  

$x\ne 0,x\ne -2$  

 

$\frac{4}{x}=\frac{3}{2+x}\mid \cdot x\left(2+x\right)$  

$4\left(2+x\right)=3x$  

$8+4x=3x\mid -4x$  

$8=-x\mid \cdot \left(-1\right)$  

$-8=x$  

---

$\text{b)}$  

Zał:   

$x+2\ne 0,x-3\ne 0$  

$x\ne -2,x\ne 3$  

 

$\frac{2}{x+2}-\frac{1}{x-3}=0\mid \cdot \left(x+2\right)\left(x-3\right)$  

$2\left(x-3\right)-\left(x+2\right)=0$  

$2x-6-x-2=0$  

$x-8=0\mid +8$  

$x=8$  

---

$\text{c)}$  

Zał:

$x-1\ne 0,x\ne 0$  

$x\ne 1,x\ne 0$  

 

$\frac{x}{x-1}=\frac{x+2}{x}\mid \cdot x\left(x-1\right)$  

$x^2=\left(x+2\right)\left(x-1\right)$  

$x^2=x^2-x+2x-2$  

$x^2=x^2+x-2\mid -x^2$  

$0=x-2\mid +2$  

$2=x$  

---

$\text{d)}$  

Zał:

$x-2\ne 0,x+2\ne 0$  

$x\ne 2,x\ne -2$  

 

$\frac{x-3}{x-2}=\frac{x+3}{x+2}\mid \cdot \left(x-2\right)\left(x+2\right)$  

$\left(x-3\right)\left(x+2\right)=\left(x+3\right)\left(x-2\right)$  

$x^2+2x-3x-6=x^2-2x+3x-6\mid -x^2+6$  

$-x=x\mid +x$  

$0=2x\mid :2$  

$0=x$  

---

$\text{e)}$   

Zał:

$x\ne 0,2x+1\ne 0$  

$x\ne 0,2x\ne -1$  

$x\ne 0,x\ne -\frac{1}{2}$  

 

$\frac{x+1}{x}=\frac{x+1}{2x+1}\mid \cdot x\left(2x+1\right)$  

$\left(x+1\right)\left(2x+1\right)=x\left(x+1\right)$  

$2x^2+x+2x+1=x^2+x\mid -x^2-x$  

$x^2+2x+1=0$  

${\left(x+1\right)}^2=0$  

$x+1=0$   

$x=-1$  

---

$\text{f)}$  

Zał:

$2x-1\ne 0,x\ne 0$  

$2x\ne 1,x\ne 0$  

$x\ne \frac{1}{2},x\ne 0$  

 

$\frac{x+1}{2x-1}-\frac{2}{x}=0\mid \cdot x\left(2x-1\right)$  

$x\left(x+1\right)-2\left(2x-1\right)=0$  

$x^2+x-4x+2=0$  

$x^2-3x+2=0$  

$\Delta ={\left(-3\right)}^2-4\cdot 1\cdot 2=9-8=1$  

$x_1=\frac{3-1}{2}=\frac{2}{2}=1$  

$x_2=\frac{3+1}{2}=\frac{4}{2}=2$