Solve for x : 1/x + 2/(2x − 3) = 1/(x − 2)
Question
Solve for x:
\[ \frac{1}{x}+\frac{2}{2x-3}=\frac{1}{x-2}, \qquad x\ne0,\frac{3}{2},2 \]Solution
Multiplying both sides by \(x(2x-3)(x-2)\),
\[ (2x-3)(x-2)+2x(x-2)=x(2x-3) \]
\[ 2x^2-7x+6+2x^2-4x=2x^2-3x \]
\[ 2x^2-8x+6=0 \]
\[ x^2-4x+3=0 \]
\[ (x-1)(x-3)=0 \]
\[ x=1 \]
or
\[ x=3 \]
Both values satisfy the given restrictions.
Answer
\[
\boxed{x=1 \quad \text{or} \quad x=3}
\]