Final answer:
The expressions (x-2)(x+6)/7(x+6) and x-2/7 are equal for all real numbers except x = -6, where the expression is undefined due to division by zero.
Step-by-step explanation:
The student is asking about the simplification of a rational expression and when the two expressions are equal. In the given expression, (x-2)(x+6)/7(x+6), we can simplify it by canceling out the common factor (x+6) in the numerator and the denominator, which leaves us with x-2/7. However, this simplification is only valid when (x+6) is not equal to zero because division by zero is undefined. Therefore, the values of x for which the two expressions are equal is all real numbers except x = -6.
To verify that the expressions are equivalent, one can check the simplified expression and the original expression for a variety of values of x, excluding x = -6. By substituting these values, one should find that the two expressions yield the same result, confirming the equality of the expressions for those values of x.