Questions: Solve: 1/(x-1)-1/(x-3)=1/(x^2-4x+3) x=

Solve: 1/(x-1)-1/(x-3)=1/(x^2-4x+3) x=
Transcript text: Solve: $\frac{1}{x-1}-\frac{1}{x-3}=\frac{1}{x^{2}-4 x+3}$ $x=$ $\square$ (For multiple answers separate with a comma.
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Solution

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Solution Steps

To solve the equation \(\frac{1}{x-1} - \frac{1}{x-3} = \frac{1}{x^2 - 4x + 3}\), we need to simplify and solve for \(x\). First, recognize that \(x^2 - 4x + 3\) can be factored into \((x-1)(x-3)\). Then, combine the fractions on the left-hand side over a common denominator and equate it to the right-hand side. Finally, solve the resulting equation for \(x\).

Step 1: Recognize the Factored Form

The given equation is: \[ \frac{1}{x-1} - \frac{1}{x-3} = \frac{1}{x^2 - 4x + 3} \] Notice that \(x^2 - 4x + 3\) can be factored as \((x-1)(x-3)\).

Step 2: Combine Fractions on the Left-Hand Side

Combine the fractions on the left-hand side over a common denominator: \[ \frac{(x-3) - (x-1)}{(x-1)(x-3)} = \frac{1}{(x-1)(x-3)} \] Simplify the numerator: \[ \frac{x-3 - x + 1}{(x-1)(x-3)} = \frac{-2}{(x-1)(x-3)} \]

Step 3: Equate and Solve

Equate the simplified left-hand side to the right-hand side: \[ \frac{-2}{(x-1)(x-3)} = \frac{1}{(x-1)(x-3)} \] Since the denominators are the same, equate the numerators: \[ -2 = 1 \] This is a contradiction, meaning there are no values of \(x\) that satisfy the equation.

Final Answer

\[ \boxed{\text{DNE}} \]

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