Final answer:
The correct expression equivalent to 30 ÷ (3 + x) is as given, without any further simplification. Incorrect distribution of division over addition is a common mistake and does not yield an equivalent expression.
Step-by-step explanation:
The student is asking which expression is equivalent to 30 ÷ (3 + x). In mathematics, when you have an expression of the form a ÷ (b + c), it is generally not equivalent to a ÷ b + a ÷ c due to the order of operations, which dictates that operations within parentheses should be done first.
Applying the rule of algebra concerning the division of a number by a binomial, we must divide the entire number by the whole binomial. Therefore, the expression 30 ÷ 3 + 30 ÷ x incorrectly distributes the division over addition, which is not a proper algebraic operation, unlike multiplication. None of the provided options correctly represents the original expression with the exception of the initial one. It's important to remember that multiplication or division should apply to every term within parentheses, and changing this can lead to a different mathematical expression.
The concept of inversion is also mentioned, which refers to a number or an algebraic expression being raised to a negative exponent. It's defined that x-n is the same as 1/xn, This underscores the importance of understanding the rules of exponents in algebra to manipulate expressions accurately.