Question

are the two expressions shown below equivalent? explain.
4(n + 3) - (3 + n) and 3n + 9

Explanation:

Step1: Expand the first expression

First, expand \(4(n + 3)\) using the distributive property \(a(b + c)=ab+ac\). So \(4(n + 3)=4n+12\). Then, we have the expression \(4(n + 3)-(3 + n)=4n + 12-(n + 3)\).

Step2: Simplify the subtraction

Remove the parentheses: \(4n+12 - n - 3\) (because when we subtract \((n + 3)\), we distribute the negative sign to both terms inside the parentheses).

Step3: Combine like terms

Combine the \(n\) terms: \(4n - n=3n\), and combine the constant terms: \(12-3 = 9\). So the simplified form of \(4(n + 3)-(3 + n)\) is \(3n+9\), which is the same as the second expression.

Answer:

Yes, the two expressions are equivalent. By expanding and simplifying \(4(n + 3)-(3 + n)\), we get \(3n + 9\), which matches the second expression.