Let's solve this step-by-step.
Given the rational expression:
[tex]\[
\frac{(x-2)(x+6)}{7(x+6)} = \frac{x-2}{7}
\][/tex]
We want to find the values of [tex]\( x \)[/tex] for which these expressions are equal.
1. Simplify the Expression:
We notice that (x+6) is a common factor in the numerator and the denominator of the left side of the equation. Thus, we can simplify the expression by canceling out the common factor (x+6), given that [tex]\( x \neq -6 \)[/tex] (since division by zero is undefined).
So,
[tex]\[
\frac{(x-2)(x+6)}{7(x+6)} = \frac{x-2}{7} \quad \text{for} \quad x \neq -6.
\][/tex]
2. Verify Simplification:
After canceling, we have:
[tex]\[
\frac{x-2}{7}.
\][/tex]
Now both sides of the equation are equal:
[tex]\[
\frac{x-2}{7} = \frac{x-2}{7}.
\][/tex]
These expressions are indeed equal for all [tex]\( x \)[/tex].
3. Identify Restrictions:
The only restriction comes from the cancellation step. The denominator [tex]\( x+6 \)[/tex] must not be zero, as division by zero is not allowed.
Therefore, setting the denominator equal to zero and solving for [tex]\( x \)[/tex] gives:
[tex]\[
x + 6 = 0 \implies x = -6.
\][/tex]
So, [tex]\( x = -6 \)[/tex] is the value that makes the denominator zero and thus is not allowed.
Therefore, the two expressions are equal for all real numbers except [tex]\( x = -6 \)[/tex].
Thus, the correct answer is:
[tex]\[ \boxed{\text{B. All real numbers except -6}} \][/tex]
