Question

solve the following rational equation.
\\(\frac{2}{x - 2}-2 = \frac{8}{x - 2}\\)

Explanation:

Step1: Eliminate the denominator

Multiply both sides of the equation \(\frac{2}{x - 2}-2=\frac{8}{x - 2}\) by \(x - 2\) (note that \(x
eq2\) to avoid division by zero) to get:
\(2-2(x - 2)=8\)

Step2: Simplify the left - hand side

Expand the left - hand side: \(2-2x + 4=8\)
Combine like terms: \(6-2x=8\)

Step3: Solve for x

Subtract 6 from both sides: \(-2x=8 - 6=2\)
Divide both sides by \(-2\): \(x=\frac{2}{-2}=-1\)

Step4: Check the solution

Substitute \(x = - 1\) into the original equation.
Left - hand side: \(\frac{2}{-1-2}-2=\frac{2}{-3}-2=-\frac{2}{3}-\frac{6}{3}=-\frac{8}{3}\)
Right - hand side: \(\frac{8}{-1 - 2}=\frac{8}{-3}=-\frac{8}{3}\)
Since the left - hand side equals the right - hand side when \(x=-1\) and \(x=-1
eq2\), the solution is valid.

Answer:

\(x=-1\)