$a\text{)}\frac{2+5x}{3}-\frac{3x+9}{3}$    

Dziedzina:

$x\in \mathbb{R}$  

$\frac{2+5x}{3}-\frac{3x+9}{3}=\frac{2+5x-3x-9}{3}=\frac{2x-7}{3}$  

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$b\text{)}\frac{\left(x-1\right)\left(x+1\right)}{2}+\frac{6-x^2}{4}$  

Dziedzina:

$x\in \mathbb{R}$  

$\frac{\left(x-1\right)\left(x+1\right)}{2}+\frac{6-x^2}{4}=\frac{2\left(x-1\right)\left(x+1\right)}{4}+\frac{6-x^2}{4}=$  

$=\frac{2\left(x^2-1\right)+6-x^2}{4}=\frac{2x^2-2+6-x^2}{4}=\frac{x^2+4}{4}$  

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$c\text{)}\frac{\left(x-2\right)x}{3}-\frac{{\left(x-1\right)}^2}{2}+\frac{5}{x}$  

Założenie:

$x\ne 0$  

Dziedzina:

$x\in \mathbb{R}-\left\{0\right\}$  

$\frac{\left(x-2\right)x}{3}-\frac{{\left(x-1\right)}^2}{2}+\frac{5}{x}=$  

$=\frac{\left(x-2\right)x\cdot 2x}{3\cdot 2\cdot x}-\frac{{\left(x-1\right)}^2\cdot 3x}{3\cdot 2\cdot x}+\frac{5\cdot 3\cdot 2}{3\cdot 2\cdot x}=$  

$=\frac{\left(x-2\right)x\cdot 2x}{6x}-\frac{{\left(x-1\right)}^2\cdot 3x}{6x}+\frac{30}{6x}=$  

$=\frac{2x^3-4x^2}{6x}-\frac{\left(x^2-2x+1\right)\cdot 3x}{6x}+\frac{30}{6x}=$  

$=\frac{2x^3-4x^2}{6x}-\frac{3x^3-6x^2+3x}{6x}+\frac{30}{6x}=$  

$=\frac{2x^3-4x^2-3x^3+6x^2-3x+30}{6x}=$  

$=\frac{-x^3+2x^2-3x+30}{6x}$  

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$d\text{)}5-\frac{x}{x+1}$  

Założenie:

$x+1\ne 0$  

$x\ne -1$  

Dziedzina:

$x\in \mathbb{R}-\left\{-1\right\}$  

$5-\frac{x}{x+1}=\frac{5\left(x+1\right)}{x+1}-\frac{x}{x+1}=\frac{5\left(x+1\right)-x}{x+1}=$  

$=\frac{5x+5-x}{x+1}=\frac{4x+5}{x+1}$  

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$e\text{)}\frac{5}{2-x}+2x$  

Założenie:

$2-x\ne 0$  

$x\ne 2$  

Dziedzina:

$x\in \mathbb{R}-\left\{2\right\}$  

$\frac{5}{2-x}+2x=\frac{5+2x\left(2-x\right)}{2-x}=\frac{5+4x-2x^2}{2-x}=$  

$=\frac{-2x^2+4x+5}{2-x}=\frac{2x^2-4x-5}{x-2}$  

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$f\text{)}\frac{6-x}{x-1}-3$  

Założenie:

$x-1\ne 0$  

$x\ne 1$  

Dziedzina:

$x\in \mathbb{R}-\left\{1\right\}$  

$\frac{6-x}{x-1}-3=\frac{6-x}{x-1}-\frac{3\left(x-1\right)}{x-1}=\frac{6-x-3\left(x-1\right)}{x-1}=$  

$=\frac{6-x-3x+3}{x-1}=\frac{9-4x}{x-1}$