Part a)
Explain the error in this simplification.
Given the simplified expression
[tex]1-\frac{2}{x-2}=\frac{x+1}{x+2}[/tex]
[tex]1-2\left(x+2\right)=\left(x+1\right)\left(x-2\right)[/tex]
[tex]\:1-2x-4=x^2-x-2[/tex]
[tex]-2x-3=x^2-x-2\:\:[/tex]
[tex]0=x^2-2x-2\:\:[/tex]
[tex]0=\left(x-1\right)\left(x-1\right)[/tex]
[tex]x=1[/tex]
Identifying the Main Error
[tex]1-\frac{2}{x-2}=\frac{x+1}{x+2}[/tex]
[tex]1-2\left(x+2\right)=\left(x+1\right)\left(x-2\right)[/tex] ← ERROR Starts here
Here is the Explanation of the Error
[tex]\mathrm{The\:equation\:should\:have\:been\:Multiplied\:by\:LCM=}\left(x-2\right)\left(x+2\right)[/tex]. In your case you wrongly multiply the equation.
CORRECTION
HERE IS HOW YOU SHOULD HAVE MULTIPLIED BY LCM = (x-2)(x+2):
[tex]1-\frac{2}{x-2}=\frac{x+1}{x+2}[/tex]
[tex]1\cdot \left(x-2\right)\left(x+2\right)-\frac{2}{x-2}\left(x-2\right)\left(x+2\right)=\frac{x+1}{x+2}\left(x-2\right)\left(x+2\right)[/tex]
[tex]\left(x-2\right)\left(x+2\right)-2\left(x+2\right)=\left(x+1\right)\left(x-2\right)[/tex]
Part b)
Show your work as you correct the error
Here is the complete correction of the error.
Considering the expression
[tex]1-\frac{2}{x-2}=\frac{x+1}{x+2}[/tex]
[tex]\mathrm{Find\:Least\:Common\:Multiplier\:of\:}x-2,\:x+2:\quad \left(x-2\right)\left(x+2\right)[/tex]
[tex]1\cdot \left(x-2\right)\left(x+2\right)-\frac{2}{x-2}\left(x-2\right)\left(x+2\right)=\frac{x+1}{x+2}\left(x-2\right)\left(x+2\right)[/tex]
[tex]\left(x-2\right)\left(x+2\right)-2\left(x+2\right)=\left(x+1\right)\left(x-2\right)[/tex]
[tex]x^2-2x-8=x^2-x-2[/tex]
[tex]x^2-2x-8+8=x^2-x-2+8[/tex]
[tex]x^2-2x=x^2-x+6[/tex]
[tex]-x=6[/tex]
[tex]\frac{-x}{-1}=\frac{6}{-1}[/tex]
[tex]x=-6[/tex]
