Lösung 2.2:3c
Aus Online Mathematik Brückenkurs 1
Start by rewriting the terms on the left-hand side as one term having a common denominator
| \displaystyle \begin{align}
\frac{1}{x-1}-\frac{1}{x+1} &= \frac{1}{x-1}\cdot\frac{x+1}{x+1} - \frac{1}{x+1}\cdot\frac{x-1}{x-1}\\[5pt] &= \frac{x+1}{(x-1)(x+1)} - \frac{x-1}{(x-1)(x+1)}\\[5pt] &= \frac{(x+1)-(x-1)}{(x-1)(x+1)}\\[5pt] &= \frac{2}{(x-1)(x+1)}\,\textrm{.} \end{align} |
If we also write \displaystyle 3x-3=3(x-1), the equation can be rewritten as
| \displaystyle \frac{2}{(x-1)(x+1)}\bigl(x^{2}+\tfrac{1}{2}\bigr) = \frac{6x-1}{3(x-1)}\,\textrm{.} |
Because \displaystyle x=1 cannot be a solution to the equation, the factor \displaystyle x-1 can be removed from the denominator of both sides (i.e. actually, we multiply both sides by \displaystyle x-1 and then eliminate it)
| \displaystyle \frac{2}{x+1}\bigl(x^{2}+\tfrac{1}{2}\bigr) = \frac{6x-1}{3}\,\textrm{.} |
Then, both sides are multiplied by 3 and \displaystyle x+1, so that we get an equation without any denominators
| \displaystyle 6\bigl(x^{2}+\tfrac{1}{2}\bigr) = (6x-1)(x+1)\,\textrm{.} |
Expanding both sides
| \displaystyle 6x^{2}+3=6x^{2}+5x-1\,\textrm{.} |
The x² terms cancel each other out and we obtain a first-order equation,
| \displaystyle 3=5x-1\,, |
which has the solution
| \displaystyle x=\frac{4}{5}\,\textrm{.} |
We check whether we have calculated correctly by substituting \displaystyle x=4/5 into the original equation,
| \displaystyle \begin{align}
\text{LHS} &= \biggl(\frac{1}{\frac{4}{5}-1}-\frac{1}{\frac{4}{5}+1}\biggr)\bigl( \bigl(\tfrac{4}{5}\bigr)^{2}+\tfrac{1}{2}\bigr) = \biggl(\frac{1}{-\frac{1}{5}}-\frac{1}{\frac{9}{5}}\biggr)\bigl( \tfrac{16}{25}+\tfrac{1}{2}\bigr)\\[5pt] &= \bigl(-5-\tfrac{5}{9}\bigr)\cdot\frac{16\cdot 2+25}{2\cdot 25} = -\frac{50}{9}\cdot\frac{57}{50} = -\frac{57}{9} = -\frac{19}{3}\,,\\[15pt] \text{RHS} &= \frac{6\cdot\frac{4}{5}-1}{3\cdot\frac{4}{5}-3} = \frac{\frac{24}{5}-\frac{5}{5}}{\frac{12}{5}-\frac{15}{5}} = \frac{\frac{1}{5}\cdot (24-5)}{\frac{1}{5}\cdot (12-15)} = \frac{19}{-3} = -\frac{19}{3}\,\textrm{.} \end{align} |
