# Likninger med x i nevner

På de to foregående sidene lærte du hvordan du skal bli kvitt en nevner med tall. Nå skal du lære å bli kvitt en nevner med bokstaver. Den gode nyheten er at fremgangsmåten er helt lik! For å finne fellesnevneren, ganger du sammen alle de ulike faktorene bare én gang. Vi går rett på to eksempler.

Eksempel 1

Løs likningen $\frac{2}{x}=3$

$\begin{array}{llll}\hfill \frac{2}{x}& =3\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x\cdot \frac{2}{x}& =3\cdot x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill 2& =3x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{2}{3}& =\frac{\text{3}x}{\text{3}}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill \frac{2}{3}& =x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill x& =\frac{2}{3}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

Eksempel 2

Løs likningen $\frac{x+1}{x}+2=\frac{3}{x+1}+3$ for $x$

Fellesnevneren er $x\left(x+1\right)$. Multipliser med fellesnevneren på begge sider:

$\begin{array}{llll}\hfill & \phantom{=}x\left(x+1\right)\cdot \phantom{\rule{-0.17em}{0ex}}\left(\frac{x+1}{x}+2\right)\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =x\left(x+1\right)\cdot \phantom{\rule{-0.17em}{0ex}}\left(\frac{3}{x+1}+3\right).\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $x\left(x+1\right)\cdot \phantom{\rule{-0.17em}{0ex}}\left(\frac{x+1}{x}+2\right)=x\left(x+1\right)\cdot \phantom{\rule{-0.17em}{0ex}}\left(\frac{3}{x+1}+3\right).$

Når du ganger med hvert ledd inni parentesen får du
$\begin{array}{llll}\hfill & \phantom{=}x\left(x+1\right)\cdot \frac{x+1}{x}+x\left(x+1\right)\cdot 2\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =x\left(x+1\right)\cdot \frac{3}{x+1}+x\left(x+1\right)\cdot 3.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

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

Videre kan du stryke noen faktorer:
$\begin{array}{llll}\hfill & \phantom{=}\text{x}\left(x+1\right)\frac{x+1}{\text{x}}+x\left(x+1\right)2\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =x\text{(x+1)}\frac{3}{\text{x+1}}+x\left(x+1\right)3.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 $\text{x}\left(x+1\right)\frac{x+1}{\text{x}}+x\left(x+1\right)2=x\text{(x+1)}\frac{3}{\text{x+1}}+x\left(x+1\right)3.$

Dette gir følgende uttrykk:
$\begin{array}{llll}\hfill & \phantom{=}{x}^{2}+2x+1+2{x}^{2}+2x\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & =3x+3{x}^{2}+3x.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 ${x}^{2}+2x+1+2{x}^{2}+2x=3x+3{x}^{2}+3x.$

Isoler alle variablene på den ene siden, og alle konstantene på den andre:
$\begin{array}{llll}\hfill -1={x}^{2}& +2{x}^{2}-3{x}^{2}\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\\ \hfill & +2x+2x-3x-3x.\phantom{\rule{2em}{0ex}}& \hfill & \phantom{\rule{2em}{0ex}}\end{array}$

 ${x}^{2}+2{x}^{2}-3{x}^{2}+2x+2x-3x-3x=-1.$

Dette kan forenkles til
 $-2x=-1.$
 $\frac{\text{−2}x}{\text{−2}}=\frac{-1}{-2}=\frac{1}{2}.$

Dermed er $x=\frac{1}{2}$.

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