Question

solve the equation.
\\(\frac{6}{x - 1} = \frac{12}{x^2 - 1}\\)

select the correct choice below and, if necessary, fill in the answer box to complete your choice.
\\(\bigcirc\\) a. \\(x = \square\\) (use a comma to separate answers as needed.)
\\(\bigcirc\\) b. there is no solution.

Explanation:

Step1: Factor the denominator

Notice that \(x^2 - 1\) is a difference of squares, so \(x^2 - 1=(x - 1)(x + 1)\). The equation becomes \(\frac{6}{x - 1}=\frac{12}{(x - 1)(x + 1)}\).

Step2: Exclude undefined values

The denominators cannot be zero, so \(x
eq1\) and \(x
eq - 1\).

Step3: Cross - multiply

Multiply both sides by \((x - 1)(x + 1)\) (since \(x
eq1\) and \(x
eq - 1\), we can do this) to get \(6(x + 1)=12\).

Step4: Solve for x

First, divide both sides by 6: \(x + 1 = 2\). Then subtract 1 from both sides: \(x=2 - 1=1\). But we know from Step 2 that \(x
eq1\) (because when \(x = 1\), the original denominator \(x - 1=0\)). So this solution is extraneous.

Answer:

B. There is no solution.