Explore Rational Equation Domains: Find the Area of (x/(5x-69) = 2/(x-1)

To solve the problem, follow these steps:

• Step 1: Identify where each denominator is zero to find the domain restrictions.

• Step 2: Solve each condition separately to exclude the non-permissible $x$ values.

Now, let's work through each step:

Step 1: The first expression involves the denominator $5x - 6$. Set it to zero:

$5x - 6 = 0$

Solve for $x$:
$5x = 6$
$x = \frac{6}{5} = 1\frac{1}{5}$

This means the function is undefined for $x = 1\frac{1}{5}$.

Step 2: The second expression involves the denominator $x - 1$. Set it to zero:

$x - 1 = 0$

Solve for $x$:
$x = 1$

This means the function is undefined for $x = 1$.

The domain of this expression is all real numbers except where these denominators are zero. Therefore, the domain restriction is:

The values of $x$ cannot equal 1 or $1\frac{1}{5}$, which corresponds to choice 3.

Therefore, the solution to the problem is $x \neq 1, x \neq 1\frac{1}{5}$.