Lösung Bruchgleichungen

Löse unter Angabe der Definitionsmenge folgende Gleichungen:

1. \(\frac{10}{x}=5 \) 

\(D = \mathbb{R} \backslash \{0\}\) 
\(L = \{2\}\)

2. \(\frac{10}{x+1}=5 \) 

\(D = \mathbb{R} \backslash \{-1\}\) 
\(L = \{1\}\)

3. \(\frac{10}{x+1}=\frac{5}{x-1} \)   

\(D = \mathbb{R} \backslash \{-1;1\}\) 
\(L = \{3\}\)

4. \(\frac{10}{x+1}=\frac{5x}{x-1}-\frac{5x^2}{x^2-1} \)  

\(D = \mathbb{R} \backslash \{-1;1\}\) 
\(\frac{10}{x+1}=\frac{5x}{x-1}-\frac{5x^2}{x^2-1} \)  \(\quad HN =(x-1)(x+1)\)
\(=\frac{10(x-1)}{(x+1)(x-1)}=\frac{5x(x+1)}{(x-1)(x+1)}-\frac{5x^2}{x^2-1} |\cdot HN \)
\(=10(x-1)=5x(x+1)-5x^2 \)
\(=10x-10=5x^2+5x-5x^2 \)
\(=5x=10 \)
\(=x=2 \)
\(L = \{2\}\)

5. \(\frac{10}{2x+2}=\frac{5}{x+1}-1 \)  

\(D = \mathbb{R} \backslash \{-1\}; \quad HN = 2(x+1)=2x+2\) 
\(10 = 10 - 2x - 2\)
\(x = -1\)
\(L = \{\} \qquad da -1 \notin D \)