\(\dfrac{x+1}{x-2}+1=\dfrac{2x}{x+2}\)
\(\Leftrightarrow\dfrac{\left(x+1\right)\left(x+2\right)}{\left(x-2\right)\left(x+2\right)}+\dfrac{\left(x+2\right)\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}=\dfrac{2x\left(x-2\right)}{\left(x+2\right)\left(x-2\right)}\)
\(\Rightarrow\left(x+1\right)\left(x+2\right)+\left(x+2\right)\left(x-2\right)=2x\left(x-2\right)\)
\(\Rightarrow x^2+3x+2+x^2-4=2x^2-4\)
\(\Rightarrow x^2+3x+2+x^2-4-2x^2+4=0\)
\(\Rightarrow3x+2=0\)
\(\Rightarrow3x=-2\)
\(\Rightarrow x=-\dfrac{2}{3}\)
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