To identify the very first error in the provided expression and steps, let's analyze it systematically.
The original equation given is:
\[ \frac{3}{x + 2} - \frac{5}{x - 6} = \frac{2x - 1}{x^2 - 4x - 12} \]
First, we need to factor the denominator on the right side. The expression \(x^2 - 4x - 12\) can be factored as follows:
\[ x^2 - 4x - 12 = (x + 2)(x - 6) \]
Now, the expression on the right-hand side becomes:
\[ \frac{2x - 1}{(x + 2)(x - 6)} \]
Next, we will rewrite the left side of the equation to have a common denominator, which is also \((x + 2)(x - 6)\):
\[ \frac{3(x - 6) - 5(x + 2)}{(x + 2)(x - 6)} = \frac{2x - 1}{(x + 2)(x - 6)} \]
Now, let's look at the first step provided:
- Step 1: \[ \frac{3(x - 6)}{(x + 2)(x - 6)} - \frac{5(x + 2)}{(a - 6)(x + 2)} = \frac{3x - 1}{(x + 2)(x - 6)} \]
In Step 1, we see a notation error—it's stated as \((a - 6)\) instead of \((x - 6)\) when it is referring to the denominator of the second term. This is the very first error in the manipulation of the expression.
Everything else appears consistent up until this point, but the introduction of \((a - 6)\) is incorrect.
Thus, the very first error occurs in Step 1, with the erroneous use of \((a - 6)\) instead of \((x - 6)\).