Solve \((x-1)/(x-2) + (x-3)/(x-4) = 3\frac{1}{3}\) for x
Question
Solve for x:
\[ \frac{x-1}{x-2}+\frac{x-3}{x-4}=3\frac{1}{3}, \qquad x\ne2,4 \]Solution
\[ \frac{x-1}{x-2}+\frac{x-3}{x-4}=\frac{10}{3} \]
Multiplying both sides by \(3(x-2)(x-4)\),
\[ 3(x-1)(x-4)+3(x-3)(x-2) = 10(x-2)(x-4) \]
\[ 3(x^2-5x+4)+3(x^2-5x+6) = 10(x^2-6x+8) \]
\[ 6x^2-30x+30 = 10x^2-60x+80 \]
\[ 4x^2-30x+50=0 \]
\[ 2x^2-15x+25=0 \]
\[ (2x-5)(x-5)=0 \]
\[ x=\frac{5}{2} \]
or
\[ x=5 \]
Both values satisfy \(x\ne2,4\).
Answer
\[
\boxed{x=\frac{5}{2}\quad \text{or}\quad x=5}
\]